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<a NAME="E1:39_1_1"/><i><font color="Green">E17</font></i>: 
1 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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="E1:39_1_1_1_1"/>
( <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> or  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_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="E1:39_1_1_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:39_1_1_1_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_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="E1:39_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> );<br/><a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 <b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:39_1_1_1_1_1_1_1_1_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="ref" href="finseq_8.html#T7" target="_self">Th7</a>);<br/></i><b>then consider </b><font color="Maroon">k0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:39_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k0</font> &amp; <font color="Maroon">k0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font> ) )
 ;<br/><a NAME="E6:39_1_1_1_1_1_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k0</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_8.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E7:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k0</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">k0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E9:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k0</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E10:39_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">k0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E11:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">k0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><a NAME="E14:39_1_1_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E15:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E16:39_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
1 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E17:39_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E19:39_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="finseq_8.html#E3:39_1_1_1_1_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E20:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>then </b><i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k0</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/> <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</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"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k0</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="finseq_8.html#E4:39_1_1_1_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, , <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
;<br/><a NAME="E23:39_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i><a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, , , </i>;<br/><a NAME="E24:39_1_1_1_1_1_1_1_1_1_1_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 ;<br/>

<a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="finseq_8.html#E1:39_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E4:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E5:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="finseq_8.html#E9:39_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E7:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_8.html#E3:39_1_1_1_1_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/> <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</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"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_8.html#E9:39_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="finseq_8.html#E13:39_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
;<br/><b>hence </b><a NAME="E9:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, <a class="txt" href="finseq_8.html#E1:39_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="finseq_8.html#E2:39_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="finseq_8.html#E5:39_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_1_1_1_1_1_1_1_1_1_1_3"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E25:39_1_1_1_1_1_1_1_1_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) ) ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 <b>by </b><i><a class="ref" href="finseq_8.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,1
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">n2</font> = 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="finseq_8.html#T7" target="_self">Th7</a>, </i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> );<br/><a NAME="E6:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E67</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 <b>by </b><i><a class="ref" href="finseq_8.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E7:39_1_1_1_1_1_1_1_1_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>);<br/></i><b>then consider </b><font color="Maroon">k0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k0</font> &amp; <font color="Maroon">k0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font> ) )
 ;<br/><a NAME="E10:39_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">k0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E11:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k0</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E12:39_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">k0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E13:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">k0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><a NAME="E16:39_1_1_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E17:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E18:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E72</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E20:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E21:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>then </b><i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k0</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="finseq_8.html#E4:39_1_1_1_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/> <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</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"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k0</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
;<br/><a NAME="E24:39_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i>, , , <a class="txt" href="finseq_8.html#E3:39_1_1_1_1_1_1_1_1"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E25:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_2_3"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 ;<br/>

<a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="finseq_8.html#E4:39_1_1_1_1_1_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>then </b><i><font color="Green">E82</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="finseq_8.html#E4:39_1_1_1_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E6:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/> <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</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"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_8.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
;<br/><b>hence </b><a NAME="E8:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_2_3_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_1_1_1_1_1_1_1_1_1_2_3"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E26:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k0</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E27:39_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1_1_1_1_1_1_1_1_2_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 ;<br/>

<a NAME="E2:39_1_1_1_1_1_1_1_1_1_1_2_4"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_1_1_1_1_1_1_1_1_2_4"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k0</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E28:39_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">k0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E29:39_1_1_1_1_1_1_1_1_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) ) ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:39_1_1_1_1_1_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) ) ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_1_1_1_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">nn</font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">n</font>,1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><a NAME="E3:39_1_1_1_1_1_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">nn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E4:39_1_1_1_1_1_1_1_2"/>
<font color="Maroon">nn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E5:39_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">nn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E6:39_1_1_1_1_1_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">nn</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:39_1_1_1_1_1_1_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">nn</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1_1_1_1_1_2_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 ;<br/>

<a NAME="E2:39_1_1_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">nn</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T7" target="_self">Th7</a>, <a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:39_1_1_1_1_1_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) ) ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 <b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_8.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_8.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_1_1_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) ) ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_1_2"/>
not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) ) ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) ) ) &amp; ( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 ;<br/>


</div>
