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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</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">F</font>, <font color="Olive" title="b2">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">G</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">G</font> )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">I</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">I</font> &amp; <font color="Olive" title="b1">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">I</font>;<br/>
<a NAME="E1:59"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</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="c2">F</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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"> ( (  for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">I</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">I</font> &amp; <font color="Olive" title="b1">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">I</font> ) implies  for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">I</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">I</font> &amp; <font color="Olive" title="b1">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">I</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">I</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">I</font> &amp; <font color="Olive" title="b1">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">I</font>
 ; <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">H</font>, <font color="Olive" title="b2">I</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">I</font> &amp; <font color="Olive" title="b1">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">I</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">H</font>, <font color="Maroon" title="c6">I</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</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">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c5">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c6">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c6">I</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:59_1"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:59_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c5">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E5:59_1"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c6">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c6">I</font></span><br/><a NAME="E6:59_1"/>
<font color="Maroon" title="c4">k</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></span>;<br/><a NAME="E7:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c4">k</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">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <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">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:59_1"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c7">x</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="c6">I</font>
 <span class="kw">and </span><br/><a NAME="E11:59_1"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</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="c8">v</font> = <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="funct_1.html#T102" title="FUNCT_1:th.102">FUNCT_1:102</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c7">x</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="#E10:59_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a></span>;<br/><a NAME="E14:59_1"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">H</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">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="finseq_1.html#T48" title="FINSEQ_1:th.48">FINSEQ_1:48</a></span>;<br/><a NAME="E15:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E10:59_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E16:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A12</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</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="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">m2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E18:59_1"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</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#T10" title="NAT_1:th.10">NAT_1:10</a></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>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">m2</font> = <font color="Maroon" title="c10">m2</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="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E20:59_1"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m2</font> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">q2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E22:59_1"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m2</font>
 <span class="kw">and </span><br/><a NAME="E23:59_1"/><span class="lab"><font color="Green" title="E53">A16</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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="#E20:59_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>);<br/></span><a NAME="E24:59_1"/><span class="lab"><font color="Green" title="E54">A17</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">q2</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">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="c13">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="c12">q2</font> or <font color="Maroon" title="c13">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">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1_1"/>
<font color="Maroon" title="c13">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="c12">q2</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="c13">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">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_1"/><span class="lab"><font color="Green" title="E55">A18</font></span>: 
<font color="Maroon" title="c14">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="c12">q2</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1_1"/><span class="lab"><font color="Green" title="E56">A19</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c15">y</font> = <font color="Maroon" title="c14">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="#E3:59_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a></span>;<br/>
<a NAME="E6:59_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E7:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A20</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">y</font>
 
<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="E8:59_1_1"/>
<font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:59_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c6">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E18:59_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E10:59_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c6">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_1_1"><span class="lab"><font color="Green" title="E57">A20</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><span class="kw">reconsider </span><font color="Maroon" title="c16">xx</font> = <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">y</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T102" title="FUNCT_1:th.102">FUNCT_1:102</a></span>;<br/>
<a NAME="E12:59_1_1"/>
<font color="Maroon" title="c16">xx</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">V</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E13:59_1_1"/>
<font color="Maroon" title="c13">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">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E23:59_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E56">A19</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><span class="kw">reconsider </span><font color="Maroon" title="c13">p</font> = <font color="Maroon" title="c5">H</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">k</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</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/><a NAME="E26:59_1"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
<font color="Maroon" title="c13">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="funct_1.html#T52" title="FUNCT_1:th.52">FUNCT_1:52</a></span>;<br/><a NAME="E27:59_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T10" title="FINSEQ_1:th.10">FINSEQ_1:10</a></span>;<br/><a NAME="E28:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A22</font></span>: 
<font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</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">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> (<font color="Maroon" title="c5">H</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="funct_1.html#T64" title="FUNCT_1:th.64">FUNCT_1:64</a></span>;<br/><a NAME="E29:59_1"/><span class="lab"><font color="Green" title="E57">A23</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E31:59_1"/><span class="lab"><font color="Green" title="E58">A24</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</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/><span class="kw">reconsider </span><font color="Maroon" title="c15">q1</font> = <font color="Maroon" title="c6">I</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</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">reconsider </span><font color="Maroon" title="c16">q2</font> = <font color="Maroon" title="c12">q2</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:59_1"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E34:59_1"/>
<font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:59_1"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E35:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A25</font></span>: 
<font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59_1"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E36:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E61">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">j</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">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="c16">q2</font> implies <font color="Maroon" title="c6">I</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">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_2"/><span class="lab"><font color="Green" title="E62">A28</font></span>: 
<font color="Maroon" title="c17">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="c16">q2</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">I</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">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font></span><br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.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"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:59_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c9">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:59_1"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:59_1_2"/>
<font color="Maroon" title="c6">I</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">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E23:59_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E62">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><span class="kw">set </span><font color="Maroon" title="c17">q</font> = <font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font>;<br/><a NAME="E38:59_1"/><span class="lab"><font color="Green" title="E62">A29</font></span>: 
<font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:59_1"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E39:59_1"/><span class="lab"><font color="Green" title="E63">A30</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c18">y</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:59_1_3"/>
 not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:59_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A31</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E3:59_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A32</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_1_3_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></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="c15">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_3_1"/><span class="lab"><font color="Green" title="E67">A34</font></span>: 
<font color="Maroon" title="c19">y1</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="c15">q1</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1_3_1"/><span class="lab"><font color="Green" title="E68">A35</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y1</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/><a NAME="E5:59_1_3_1"/><span class="lab"><font color="Green" title="E69">A36</font></span>: 
<font color="Maroon" title="c19">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="txt" href="#E3:59_1_3_1"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">y1</font> = <font color="Maroon" title="c19">y1</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="#E3:59_1_3_1"><span class="lab"><font color="Green" title="E67">A34</font></span></a></span>;<br/><a NAME="E7:59_1_3_1"/>
<font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1_3_1"><span class="lab"><font color="Green" title="E69">A36</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E8:59_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A37</font></span>: 
<font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:59_1_3_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1_3_1"><span class="lab"><font color="Green" title="E69">A36</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E10:59_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E8:59_1_3_1"><span class="lab"><font color="Green" title="E70">A37</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E11:59_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A38</font></span>: 
<font color="Maroon" title="c20">y1</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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E12:59_1_3_1"/>
<font color="Maroon" title="c15">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_3_1"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E13:59_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A39</font></span>: 
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E3:59_1_3"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="txt" href="#E4:59_1_3_1"><span class="lab"><font color="Green" title="E68">A35</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E14:59_1_3_1"/>
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:59_1"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E5:59_1_3_1"><span class="lab"><font color="Green" title="E69">A36</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E15:59_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E10:59_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E11:59_1_3_1"><span class="lab"><font color="Green" title="E71">A38</font></span></a>, <a class="txt" href="#E13:59_1_3_1"><span class="lab"><font color="Green" title="E72">A39</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 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="E5:59_1_3"/><span class="lab"><font color="Green" title="E67">A40</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E3:59_1_3"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E68">A41</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_1_3_2"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></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="c16">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_3_2"/><span class="lab"><font color="Green" title="E69">A42</font></span>: 
<font color="Maroon" title="c19">y1</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="c16">q2</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1_3_2"/><span class="lab"><font color="Green" title="E70">A43</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y1</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="c20">y1</font> = <font color="Maroon" title="c19">y1</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="#E3:59_1_3_2"><span class="lab"><font color="Green" title="E69">A42</font></span></a></span>;<br/><a NAME="E6:59_1_3_2"/>
<font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E3:59_1_3_2"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:59_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A44</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E8:59_1_3_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:59_1"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E9:59_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E7:59_1_3_2"><span class="lab"><font color="Green" title="E71">A44</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E10:59_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A45</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E11:59_1_3_2"/>
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E23:59_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E5:59_1_3"><span class="lab"><font color="Green" title="E67">A40</font></span></a>, <a class="txt" href="#E3:59_1_3_2"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="txt" href="#E4:59_1_3_2"><span class="lab"><font color="Green" title="E70">A43</font></span></a></span>;<br/><a NAME="E12:59_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A46</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E10:59_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E10:59_1_3_2"><span class="lab"><font color="Green" title="E72">A45</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E13:59_1_3_2"/>
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E3:59_1_3_2"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E14:59_1_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59_1_3_2"><span class="lab"><font color="Green" title="E73">A46</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="E7:59_1_3"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_3"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:59_1_3"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:59_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1_3"><span class="lab"><font color="Green" title="E66">A33</font></span></a>, <a class="txt" href="#E6:59_1_3"><span class="lab"><font color="Green" title="E68">A41</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="E40:59_1"/><span class="lab"><font color="Green" title="E64">A47</font></span>: 
<font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">y1</font>, <font color="Maroon" title="c19">y2</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="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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="c18">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> or  not <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y2</font> or <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_1_4"/><span class="lab"><font color="Green" title="E65">A48</font></span>: 
( <font color="Maroon" title="c18">y1</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">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> &amp; <font color="Maroon" title="c19">y2</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">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:59_1_4"/><span class="lab"><font color="Green" title="E66">A49</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y2</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="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">x1</font> = <font color="Maroon" title="c18">y1</font>, <font color="Maroon" title="c21">x2</font> = <font color="Maroon" title="c19">y2</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="#E1:59_1_4"><span class="lab"><font color="Green" title="E65">A48</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E67">A50</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c22">j1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E2:59_1_4_1"/><span class="lab"><font color="Green" title="E68">A51</font></span>: 
<font color="Maroon" title="c22">j1</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="c16">q2</font>
 <span class="kw">and </span><br/><a NAME="E3:59_1_4_1"/><span class="lab"><font color="Green" title="E69">A52</font></span>: 
<font color="Maroon" title="c20">x1</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> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</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">j2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j2</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="c16">q2</font> &amp; <font color="Maroon" title="c21">x2</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> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j2</font> ) implies <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font> )</span><br/><a NAME="E4:59_1_4_1"/><span class="lab"><font color="Green" title="E70">A53</font></span>: 
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E23:59_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E2:59_1_4_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a></span>;<br/><a NAME="E5:59_1_4_1"/>
<font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E2:59_1_4_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:59_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A54</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E7:59_1_4_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:59_1"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E8:59_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E6:59_1_4_1"><span class="lab"><font color="Green" title="E71">A54</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:59_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A55</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c23">j2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E11:59_1_4_1"/><span class="lab"><font color="Green" title="E73">A56</font></span>: 
<font color="Maroon" title="c23">j2</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="c16">q2</font>
 <span class="kw">and </span><br/><a NAME="E12:59_1_4_1"/><span class="lab"><font color="Green" title="E74">A57</font></span>: 
<font color="Maroon" title="c21">x2</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> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</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="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font></span><br/><a NAME="E13:59_1_4_1"/><span class="lab"><font color="Green" title="E75">A58</font></span>: 
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E23:59_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E11:59_1_4_1"><span class="lab"><font color="Green" title="E73">A56</font></span></a></span>;<br/><a NAME="E14:59_1_4_1"/>
<font color="Maroon" title="c23">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E11:59_1_4_1"><span class="lab"><font color="Green" title="E73">A56</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E15:59_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A59</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E16:59_1_4_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:59_1"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E17:59_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E15:59_1_4_1"><span class="lab"><font color="Green" title="E76">A59</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E18:59_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A60</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">j2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:59_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="txt" href="#E2:59_1_4"><span class="lab"><font color="Green" title="E66">A49</font></span></a>, <a class="txt" href="#E2:59_1_4_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E3:59_1_4_1"><span class="lab"><font color="Green" title="E69">A52</font></span></a>, <a class="txt" href="#E12:59_1_4_1"><span class="lab"><font color="Green" title="E74">A57</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:59_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="txt" href="#E11:59_1_4_1"><span class="lab"><font color="Green" title="E73">A56</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/><a NAME="E20:59_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E4:59_1_4_1"><span class="lab"><font color="Green" title="E70">A53</font></span></a>, <a class="txt" href="#E13:59_1_4_1"><span class="lab"><font color="Green" title="E75">A58</font></span></a>, <a class="txt" href="#E9:59_1_4_1"><span class="lab"><font color="Green" title="E72">A55</font></span></a>, <a class="txt" href="#E18:59_1_4_1"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E21:59_1_4_1"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_4_1"><span class="lab"><font color="Green" title="E69">A52</font></span></a>, <a class="txt" href="#E12:59_1_4_1"><span class="lab"><font color="Green" title="E74">A57</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>
<div><span class="lab"><font color="Green" title="E68">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_1_4_2"/><span class="lab"><font color="Green" title="E69">A62</font></span>: 
<font color="Maroon" title="c21">x2</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="c15">q1</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">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c16">q2</font> &amp; <font color="Maroon" title="c20">x1</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> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> ) implies <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font> )</span><br/><a NAME="E2:59_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E3:59_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A63</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_4_2"><span class="lab"><font color="Green" title="E69">A62</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E4:59_1_4_2"/><span class="lab"><font color="Green" title="E71">A64</font></span>: 
<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="txt" href="#E1:59_1_4_2"><span class="lab"><font color="Green" title="E69">A62</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E5:59_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A65</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:59_1_4_2"/><span class="lab"><font color="Green" title="E73">A66</font></span>: 
<font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1_4_2"><span class="lab"><font color="Green" title="E71">A64</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:59_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:59_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E5:59_1_4_2"><span class="lab"><font color="Green" title="E72">A65</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:59_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A67</font></span>: 
<font color="Maroon" title="c21">x2</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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c22">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E11:59_1_4_2"/><span class="lab"><font color="Green" title="E75">A68</font></span>: 
<font color="Maroon" title="c22">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="c16">q2</font>
 <span class="kw">and </span><br/><a NAME="E12:59_1_4_2"/><span class="lab"><font color="Green" title="E76">A69</font></span>: 
<font color="Maroon" title="c20">x1</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> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</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="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font></span><br/><a NAME="E13:59_1_4_2"/>
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E23:59_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E11:59_1_4_2"><span class="lab"><font color="Green" title="E75">A68</font></span></a></span>;<br/><a NAME="E14:59_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A70</font></span>: 
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_4"><span class="lab"><font color="Green" title="E66">A49</font></span></a>, <a class="txt" href="#E11:59_1_4_2"><span class="lab"><font color="Green" title="E75">A68</font></span></a>, <a class="txt" href="#E12:59_1_4_2"><span class="lab"><font color="Green" title="E76">A69</font></span></a>, <a class="txt" href="#E3:59_1_4_2"><span class="lab"><font color="Green" title="E70">A63</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E15:59_1_4_2"/>
<font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E11:59_1_4_2"><span class="lab"><font color="Green" title="E75">A68</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E16:59_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A71</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E17:59_1_4_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:59_1"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E18:59_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E16:59_1_4_2"><span class="lab"><font color="Green" title="E78">A71</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E19:59_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A72</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E20:59_1_4_2"/><span class="lab"><font color="Green" title="E80">A73</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 <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="E21:59_1_4_2"/>
<font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:59_1"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E6:59_1_4_2"><span class="lab"><font color="Green" title="E73">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E22:59_1_4_2"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E14:59_1_4_2"><span class="lab"><font color="Green" title="E77">A70</font></span></a>, <a class="txt" href="#E9:59_1_4_2"><span class="lab"><font color="Green" title="E74">A67</font></span></a>, <a class="txt" href="#E19:59_1_4_2"><span class="lab"><font color="Green" title="E79">A72</font></span></a>, <a class="txt" href="#E20:59_1_4_2"><span class="lab"><font color="Green" title="E80">A73</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 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>
<div><span class="lab"><font color="Green" title="E69">A74</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_1_4_3"/><span class="lab"><font color="Green" title="E70">A75</font></span>: 
<font color="Maroon" title="c20">x1</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="c15">q1</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">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c16">q2</font> &amp; <font color="Maroon" title="c21">x2</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> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> ) implies <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font> )</span><br/><a NAME="E2:59_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E3:59_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A76</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_4_3"><span class="lab"><font color="Green" title="E70">A75</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E4:59_1_4_3"/><span class="lab"><font color="Green" title="E72">A77</font></span>: 
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="txt" href="#E1:59_1_4_3"><span class="lab"><font color="Green" title="E70">A75</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E5:59_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A78</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:59_1_4_3"/><span class="lab"><font color="Green" title="E74">A79</font></span>: 
<font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1_4_3"><span class="lab"><font color="Green" title="E72">A77</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:59_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:59_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E5:59_1_4_3"><span class="lab"><font color="Green" title="E73">A78</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:59_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A80</font></span>: 
<font color="Maroon" title="c20">x1</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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c22">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E11:59_1_4_3"/><span class="lab"><font color="Green" title="E76">A81</font></span>: 
<font color="Maroon" title="c22">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="c16">q2</font>
 <span class="kw">and </span><br/><a NAME="E12:59_1_4_3"/><span class="lab"><font color="Green" title="E77">A82</font></span>: 
<font color="Maroon" title="c21">x2</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> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</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="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font></span><br/><a NAME="E13:59_1_4_3"/>
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E23:59_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E11:59_1_4_3"><span class="lab"><font color="Green" title="E76">A81</font></span></a></span>;<br/><a NAME="E14:59_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A83</font></span>: 
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_4"><span class="lab"><font color="Green" title="E66">A49</font></span></a>, <a class="txt" href="#E11:59_1_4_3"><span class="lab"><font color="Green" title="E76">A81</font></span></a>, <a class="txt" href="#E12:59_1_4_3"><span class="lab"><font color="Green" title="E77">A82</font></span></a>, <a class="txt" href="#E3:59_1_4_3"><span class="lab"><font color="Green" title="E71">A76</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E15:59_1_4_3"/>
<font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E11:59_1_4_3"><span class="lab"><font color="Green" title="E76">A81</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E16:59_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A84</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E17:59_1_4_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:59_1"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E18:59_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="#E16:59_1_4_3"><span class="lab"><font color="Green" title="E79">A84</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E19:59_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A85</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">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="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E20:59_1_4_3"/><span class="lab"><font color="Green" title="E81">A86</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 <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="E21:59_1_4_3"/>
<font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:59_1"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E6:59_1_4_3"><span class="lab"><font color="Green" title="E74">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E22:59_1_4_3"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E14:59_1_4_3"><span class="lab"><font color="Green" title="E78">A83</font></span></a>, <a class="txt" href="#E9:59_1_4_3"><span class="lab"><font color="Green" title="E75">A80</font></span></a>, <a class="txt" href="#E19:59_1_4_3"><span class="lab"><font color="Green" title="E80">A85</font></span></a>, <a class="txt" href="#E20:59_1_4_3"><span class="lab"><font color="Green" title="E81">A86</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 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="E7:59_1_4"/><span class="lab"><font color="Green" title="E70">A87</font></span>: 
<font color="Maroon" title="c15">q1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="funct_1.html#T52" title="FUNCT_1:th.52">FUNCT_1:52</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_1_4_4"/><span class="lab"><font color="Green" title="E71">A88</font></span>: 
( <font color="Maroon" title="c20">x1</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="c15">q1</font> &amp; <font color="Maroon" title="c21">x2</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="c15">q1</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="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font></span><br/><a NAME="E2:59_1_4_4"/><span class="kw">then </span>
( <font color="Maroon" title="c15">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> &amp; <font color="Maroon" title="c15">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:59_1_4_4"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_4"><span class="lab"><font color="Green" title="E66">A49</font></span></a>, <a class="txt" href="#E7:59_1_4"><span class="lab"><font color="Green" title="E70">A87</font></span></a>, <a class="txt" href="#E1:59_1_4_4"><span class="lab"><font color="Green" title="E71">A88</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 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>
<span class="kw">hence </span><a NAME="E9:59_1_4"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_4"><span class="lab"><font color="Green" title="E65">A48</font></span></a>, <a class="txt" href="#E6:59_1_4"><span class="lab"><font color="Green" title="E69">A74</font></span></a>, <a class="txt" href="#E5:59_1_4"><span class="lab"><font color="Green" title="E68">A61</font></span></a>, <a class="txt" href="#E4:59_1_4"><span class="lab"><font color="Green" title="E67">A50</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</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="E41:59_1"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</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="E42:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A89</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E66">A90</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> implies <font color="Maroon" title="c5">H</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="c13">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span> <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"><font color="Maroon" title="c8">v</font></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="c18">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_5"/>
<font color="Maroon" title="c18">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"><font color="Maroon" title="c8">v</font></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"> <font color="Maroon" title="c5">H</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="c13">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span> <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"><font color="Maroon" title="c8">v</font></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="c18">k</font></span><br/><a NAME="E2:59_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><a NAME="E3:59_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <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#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</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="E4:59_1_5"/>
<font color="Maroon" title="c5">H</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="c13">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span> <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"><font color="Maroon" title="c8">v</font></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="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:59_1"><span class="lab"><font color="Green" title="E65">A89</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</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><div><span class="lab"><font color="Green" title="E67">A91</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">j</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> implies <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_6"/><span class="lab"><font color="Green" title="E68">A92</font></span>: 
<font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 ; <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">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span><br/><div><span class="lab"><font color="Green" title="E69">A93</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_1_6_1"/>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</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">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span><br/><a NAME="E2:59_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A94</font></span>: 
<font color="Maroon" title="c18">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="c15">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E3:59_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_1_6_1"/>
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_6_1"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</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><div><span class="lab"><font color="Green" title="E70">A95</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_6_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:59_1_6_2"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp;  <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"><font color="Maroon" title="c8">v</font></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#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><a NAME="E2:59_1_6_2"/><span class="kw">then </span>
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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:59_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A96</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <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="c15">q1</font></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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></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> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">assume </span><a NAME="E4:59_1_6_2"/>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span><br/><a NAME="E5:59_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:59_1_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E31:59_1"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="txt" href="#E36:59_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="txt" href="#E3:59_1_6_2"><span class="lab"><font color="Green" title="E71">A96</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</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 href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.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"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:59_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
;<br/><a NAME="E5:59_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_6"><span class="lab"><font color="Green" title="E68">A92</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:59_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:59_1"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E7:59_1_6"/>
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_6"><span class="lab"><font color="Green" title="E69">A93</font></span></a>, <a class="txt" href="#E3:59_1_6"><span class="lab"><font color="Green" title="E70">A95</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="E45:59_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</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="c16">q2</font></span>)</span> = 
<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="c15">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">m2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:59_1"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="txt" href="#E18:59_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E36:59_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
;<br/><a NAME="E47:59_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E48:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A97</font></span>: 
<font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:59_1"><span class="lab"><font color="Green" title="E67">A91</font></span></a>, <a class="txt" href="#E37:59_1"><span class="lab"><font color="Green" title="E61">A27</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">then </span> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<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="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>
;<br/><a NAME="E50:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A98</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:59_1"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="xboole_1.html#T88" title="XBOOLE_1:th.88">XBOOLE_1:88</a></span>;<br/><a NAME="E51:59_1"/><span class="lab"><font color="Green" title="E70">A99</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E52:59_1"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</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">k</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T113" title="RELAT_1:th.113">RELAT_1:113</a>, <a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>;<br/><a NAME="E53:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A100</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E50:59_1"><span class="lab"><font color="Green" title="E69">A98</font></span></a>, <a class="txt" href="#E51:59_1"><span class="lab"><font color="Green" title="E70">A99</font></span></a>, <a class="txt" href="#E28:59_1"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="funct_1.html#T59" title="FUNCT_1:th.59">FUNCT_1:59</a></span>;<br/><a NAME="E54:59_1"/>
(  <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"><font color="Maroon" title="c8">v</font></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 &amp; (  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="c13">p</font> holds <br/><font color="Maroon" title="c5">H</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" title="c13">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> ) )
 <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>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E55:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="c8">v</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="#E42:59_1"><span class="lab"><font color="Green" title="E65">A89</font></span></a>, <a class="txt" href="#E51:59_1"><span class="lab"><font color="Green" title="E70">A99</font></span></a>, <a class="txt" href="#E43:59_1"><span class="lab"><font color="Green" title="E66">A90</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E56:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A101</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T41" target="_self" title="RLVECT_1:th.41">Th58</a></span>;<br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c6">I</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:59_1"><span class="lab"><font color="Green" title="E68">A97</font></span></a>, <a class="ref" href="rlvect_1.html#T41" target="_self" title="RLVECT_1:th.41">Th58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T41" target="_self" title="RLVECT_1:th.41">Th58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">q1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" target="_self" title="RLVECT_1:def.3">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T41" target="_self" title="RLVECT_1:th.41">Th58</a></span>
;<br/><span class="kw">hence </span><a NAME="E58:59_1"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E42:59_1"><span class="lab"><font color="Green" title="E65">A89</font></span></a>, <a class="txt" href="#E53:59_1"><span class="lab"><font color="Green" title="E71">A100</font></span></a>, <a class="txt" href="#E26:59_1"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E40:59_1"><span class="lab"><font color="Green" title="E64">A47</font></span></a>, <a class="txt" href="#E56:59_1"><span class="lab"><font color="Green" title="E72">A101</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="E3:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A102</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">H</font>, <font color="Maroon" title="c5">I</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</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">H</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c4">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c5">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">I</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_2"/><span class="lab"><font color="Green" title="E40">A103</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">H</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="E2:59_2"/><span class="lab"><font color="Green" title="E41">A104</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">and </span><br/><a NAME="E3:59_2"/>
<font color="Maroon" title="c4">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E4:59_2"/>
<font color="Maroon" title="c5">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">I</font></span><br/><a NAME="E5:59_2"/>
<font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E40">A103</font></span></a></span>;<br/><a NAME="E6:59_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_2"><span class="lab"><font color="Green" title="E41">A104</font></span></a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a>, <a class="ref" href="relat_1.html#T41" title="RELAT_1:th.41">RELAT_1:41</a></span>;<br/><a NAME="E7:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A105</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E8:59_2"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E40">A103</font></span></a>, <a class="txt" href="rlvect_1.html#E54"><span class="lab"><font color="Green" title="E34">Lm5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:59_2"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_2"><span class="lab"><font color="Green" title="E42">A105</font></span></a>, <a class="txt" href="rlvect_1.html#E54"><span class="lab"><font color="Green" title="E34">Lm5</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="E5:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A106</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E6:59"/>
 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="#E5:59"><span class="lab"><font color="Green" title="E40">A106</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E39">A102</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:59"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E38">A1</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>
