<?xml version="1.0"?>
<div class="add">

<span class="kw">consider </span><font color="Maroon" title="c1">r</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:147_1"/><span class="lab"><font color="Green" title="E96">A5</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E3:147_1"/><span class="lab"><font color="Green" title="E97">A6</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon">F<sub>6</sub></font>()
 <span class="kw">and </span><br/><a NAME="E4:147_1"/><span class="lab"><font color="Green" title="E98">A7</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font> holds <br/>( <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>6</sub></font>() &amp; <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="aofa_000.html#D33" target="_self" title="AOFA_000:def.33">Def33</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">r</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1</span>)</span>] );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">r</font> implies  ex <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>() st <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Maroon">F<sub>2</sub></font>() );<br/>
<a NAME="E5:147_1"/><span class="lab"><font color="Green" title="E99">A8</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_1"><span class="lab"><font color="Green" title="E96">A5</font></span></a></span>;<br/>
<a NAME="E6:147_1"/><span class="lab"><font color="Green" title="E100">A9</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:147_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>() st <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Maroon">F<sub>2</sub></font>()</span><br/>

<a NAME="E3:147_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E4:147_1_1"/><span class="lab"><font color="Green" title="E102">A11</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E5:147_1_1"/><span class="lab"><font color="Green" title="E103">A12</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_1_1"><span class="lab"><font color="Green" title="E101">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:147_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A13</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>6</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_1"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="txt" href="#E4:147_1_1"><span class="lab"><font color="Green" title="E102">A11</font></span></a></span>;<br/>
<a NAME="E7:147_1_1"/><span class="lab"><font color="Green" title="E105">A14</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_1"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="txt" href="#E4:147_1_1"><span class="lab"><font color="Green" title="E102">A11</font></span></a>, <a class="txt" href="#E5:147_1_1"><span class="lab"><font color="Green" title="E103">A12</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">q1</font> = <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_1_1"><span class="lab"><font color="Green" title="E104">A13</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">q3</font> = <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">q1</font>,<font color="Maroon">F<sub>3</sub></font>() as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>() ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">q3</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">q3</font>,<font color="Maroon">F<sub>2</sub></font>()</span><br/>

<span class="kw">thus </span><a NAME="E10:147_1_1"/>
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">q3</font>,<font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147_1_1"><span class="lab"><font color="Green" title="E105">A14</font></span></a>, <a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">Def29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:147_1"/><span class="lab"><font color="Green" title="E101">A15</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="txt" href="#E2:147_1"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="txt" href="#E3:147_1"><span class="lab"><font color="Green" title="E97">A6</font></span></a>, <a class="txt" href="#E4:147_1"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="ref" href="aofa_000.html#T86" target="_self" title="AOFA_000:th.86">Th86</a></span>;<br/>
<a NAME="E8:147_1"/><span class="lab"><font color="Green" title="E102">A16</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_1_2"/><span class="lab"><font color="Green" title="E103">A17</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:147_1_2"/><span class="lab"><font color="Green" title="E104">A18</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">j</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:147_1_2"/><span class="lab"><font color="Green" title="E105">A19</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_1_2"><span class="lab"><font color="Green" title="E104">A18</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E5:147_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:147_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></span>;<br/>
<a NAME="E7:147_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A20</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E8:147_1_2"/><span class="lab"><font color="Green" title="E107">A21</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E9:147_1_2"/><span class="lab"><font color="Green" title="E108">A22</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147_1_2"><span class="lab"><font color="Green" title="E106">A20</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E10:147_1_2"/><span class="lab"><font color="Green" title="E109">A23</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147_1_2"><span class="lab"><font color="Green" title="E106">A20</font></span></a>, <a class="txt" href="#E8:147_1_2"><span class="lab"><font color="Green" title="E107">A21</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E11:147_1_2"/><span class="lab"><font color="Green" title="E110">A24</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>6</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_1"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="txt" href="#E4:147_1_2"><span class="lab"><font color="Green" title="E105">A19</font></span></a>, <a class="txt" href="#E9:147_1_2"><span class="lab"><font color="Green" title="E108">A22</font></span></a></span>;<br/>
<a NAME="E12:147_1_2"/><span class="lab"><font color="Green" title="E111">A25</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_1"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="txt" href="#E4:147_1_2"><span class="lab"><font color="Green" title="E105">A19</font></span></a>, <a class="txt" href="#E9:147_1_2"><span class="lab"><font color="Green" title="E108">A22</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">q1</font> = <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c5">q2</font> = <font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:147_1_2"><span class="lab"><font color="Green" title="E109">A23</font></span></a>, <a class="txt" href="#E11:147_1_2"><span class="lab"><font color="Green" title="E110">A24</font></span></a>, <a class="ref" href="dtconstr.html#T2" title="DTCONSTR:th.2">DTCONSTR:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">q9</font> = <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon">F<sub>3</sub></font>() as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>() ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">j9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E16:147_1_2"/><span class="lab"><font color="Green" title="E112">A26</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_1_2"><span class="lab"><font color="Green" title="E105">A19</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<a NAME="E17:147_1_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E5:147_1"><span class="lab"><font color="Green" title="E99">A8</font></span></a>, <a class="txt" href="#E6:147_1"><span class="lab"><font color="Green" title="E100">A9</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">qq</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E19:147_1_2"/><span class="lab"><font color="Green" title="E113">A27</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j9</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">qq</font>,<font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_1_2"><span class="lab"><font color="Green" title="E104">A18</font></span></a>, <a class="txt" href="#E16:147_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a></span>;<br/>
<a NAME="E20:147_1_2"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">q9</font>,<font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:147_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">Def29</a></span>;<br/>
<a NAME="E21:147_1_2"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">qq</font>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">F<sub>3</sub></font>()]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="txt" href="#E1:147_1_2"><span class="lab"><font color="Green" title="E103">A17</font></span></a>, <a class="txt" href="#E7:147_1_2"><span class="lab"><font color="Green" title="E106">A20</font></span></a>, <a class="txt" href="#E8:147_1_2"><span class="lab"><font color="Green" title="E107">A21</font></span></a>, <a class="txt" href="#E16:147_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E19:147_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:147_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147"><span class="lab"><font color="Green" title="E94">A3</font></span></a>, <a class="txt" href="#E4:147_1"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="txt" href="#E4:147_1_2"><span class="lab"><font color="Green" title="E105">A19</font></span></a>, <a class="txt" href="#E9:147_1_2"><span class="lab"><font color="Green" title="E108">A22</font></span></a>, <a class="txt" href="#E16:147_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E19:147_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:147_1"/><span class="lab"><font color="Green" title="E103">A28</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E7:147_1"><span class="lab"><font color="Green" title="E101">A15</font></span></a>, <a class="txt" href="#E8:147_1"><span class="lab"><font color="Green" title="E102">A16</font></span></a>);<br/></span>
<a NAME="E10:147_1"/><span class="lab"><font color="Green" title="E104">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:147_1"/><span class="lab"><font color="Green" title="E105">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<a NAME="E13:147_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:147_1"><span class="lab"><font color="Green" title="E104">A29</font></span></a>, <a class="txt" href="#E12:147_1"><span class="lab"><font color="Green" title="E105">A30</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:147_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>5</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="txt" href="#E2:147_1"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="txt" href="#E9:147_1"><span class="lab"><font color="Green" title="E103">A28</font></span></a>, <a class="txt" href="#E12:147_1"><span class="lab"><font color="Green" title="E105">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
