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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F2</font> &amp; (  for <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</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="Olive" title="b1">F1</font> holds <br/><font color="Olive" title="b2">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b2">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F2</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c2">F1</font> holds <br/><font color="Maroon" title="c3">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F2</font> &amp; (  for <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</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="Olive" title="b1">F1</font> holds <br/><font color="Olive" title="b2">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b2">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> ;<br/>
<a NAME="E1:17"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c5">F1</font> holds <br/><font color="Maroon" title="c6">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:17_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c5">F1</font> holds <br/><font color="Maroon" title="c6">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c6">F2</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>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E7:17_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E8:17_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c8">q</font> holds <br/><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E9:17_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E10:17_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="finseq_3.html#T53" title="FINSEQ_3:th.53">FINSEQ_3:53</a></span>;<br/>
<a NAME="E11:17_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">q</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:17_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c10">y</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="c8">q</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a></span>;<br/>
<a NAME="E6:17_1_1"/>
<font color="Maroon" title="c11">y</font> <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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E7:17_1_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">y</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E8:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c11">y</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="c6">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:17_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a></span>;<br/>
<a NAME="E10:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E11:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:17_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:17_1"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E13:17_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <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#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E14:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</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="c5">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E15:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17_1"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/>
<a NAME="E17:17_1"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E21">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">k</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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> implies <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_1_2"/>
<font color="Maroon" title="c10">k</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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><a NAME="E2:17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A18</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/><a NAME="E3:17_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</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="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E4:17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</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="c5">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:17_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>;<br/><a NAME="E6:17_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E7:17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E4:17_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E8:17_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:17_1_2"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c8">q</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">k</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="c7">p</font> implies <font color="Maroon" title="c10">q</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:17_1_3"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
<font color="Maroon" title="c11">k</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="c7">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">q</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <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="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="group_4.html#E18"><span class="lab"><font color="Green" title="E7">Lm3</font></span></a></span>;<br/><a NAME="E3:17_1_3"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.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="c5">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></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="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a></span>;<br/><a NAME="E4:17_1_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <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="c7">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="group_4.html#E18"><span class="lab"><font color="Green" title="E7">Lm3</font></span></a></span>;<br/><a NAME="E5:17_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</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="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A23</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E9:17_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>;<br/><a NAME="E7:17_1_3"/>
<font color="Maroon" title="c11">k</font> <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="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A24</font></span>: 
<font color="Maroon" title="c11">k</font> <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="c5">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E9:17_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E10:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A25</font></span>: 
<font color="Maroon" title="c11">k</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="c5">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_1_3"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> = 
<font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c7">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:17_1_3"/>
<font color="Maroon" title="c10">q</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:17_1_3"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="txt" href="#E10:17_1_3"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E3:17_1_3"><span class="lab"><font color="Green" title="E23">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E21:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A26</font></span>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">k</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="c10">q</font> implies <font color="Maroon" title="c6">F2</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> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )</span><br/><a NAME="E1:17_1_4"/><span class="lab"><font color="Green" title="E24">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><span class="kw">assume </span><a NAME="E2:17_1_4"/>
<font color="Maroon" title="c11">k</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="c10">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F2</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> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/><span class="kw">hence </span><a NAME="E3:17_1_4"/>
<font color="Maroon" title="c6">F2</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> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E1:17_1_4"><span class="lab"><font color="Green" title="E24">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E23:17_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/>
<a NAME="E24:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:17_1"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E22:17_1"><span class="lab"><font color="Green" title="E23">A27</font></span></a>, <a class="ref" href="finseq_3.html#T38" title="FINSEQ_3:th.38">FINSEQ_3:38</a></span>;<br/>
<span class="kw">then </span> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">F2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c10">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T6" title="FINSOP_1:th.6">FINSOP_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:17_1"><span class="lab"><font color="Green" title="E22">A26</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T17" title="GROUP_1:th.17">GROUP_1:17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E26:17_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="finsop_1.html#T4" title="FINSOP_1:th.4">FINSOP_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:17"/><span class="lab"><font color="Green" title="E10">A29</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F2</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c4">F1</font> holds <br/><font color="Maroon" title="c5">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E11">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:17_2"/><span class="lab"><font color="Green" title="E12">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</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="c4">F1</font> &amp;  not <font color="Maroon" title="c5">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) or  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/>

<a NAME="E3:17_2"/>
<font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E11">A30</font></span></a></span>;<br/>
<a NAME="E4:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A32</font></span>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" target="_self" title="GROUP_4:th.8">Th11</a></span>;<br/>
<a NAME="E5:17_2"/>
<font color="Maroon" title="c5">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E2:17_2"><span class="lab"><font color="Green" title="E12">A31</font></span></a></span>;<br/>
<a NAME="E6:17_2"/><span class="kw">then </span>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" target="_self" title="GROUP_4:th.8">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:17_2"/>
(  ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</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="c4">F1</font> &amp;  not <font color="Maroon" title="c5">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) or  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_2"><span class="lab"><font color="Green" title="E13">A32</font></span></a>, <a class="ref" href="group_1.html#T8" title="GROUP_1:th.8">GROUP_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:17"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E10">A29</font></span></a>, <a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E9">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:17"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F2</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c2">F1</font> holds <br/><font color="Maroon" title="c3">F2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
