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

<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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> being   <a href="pdiff_6.html#NM1" title="PDIFF_6:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font><br/>  for <font color="Olive" title="b2">xseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><br/>  for <font color="Olive" title="b3">yseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">yseq</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">xseq</font> holds <br/><font color="Olive" title="b3">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span> ) holds <br/> <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b3">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">xseq</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="pdiff_6.html#NM1" title="PDIFF_6:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</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">xseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><br/>  for <font color="Olive" title="b2">yseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">yseq</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">xseq</font> holds <br/><font color="Olive" title="b2">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> ) holds <br/> <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b1">xseq</font></span>)</span></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">xseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><br/>  for <font color="Olive" title="b2">yseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">xseq</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">yseq</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">xseq</font> holds <br/><font color="Olive" title="b2">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> ) holds <br/> <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b1">xseq</font></span>)</span>;<br/>
<a NAME="E1:22"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">xseq</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</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">yseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c4">xseq</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">yseq</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">xseq</font> holds <br/><font color="Olive" title="b1">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) holds <br/> <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b1">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">yseq</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c4">xseq</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">yseq</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">xseq</font> holds <br/><font color="Maroon" title="c5">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:22_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c4">xseq</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">yseq</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">xseq</font> holds <br/><font color="Maroon" title="c5">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span></span><br/>

<a NAME="E2:22_1"/><span class="kw">then </span>
(  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D11" title="EUCLID_7:def.11">EUCLID_7:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:22_1"/>
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#T11" target="_self" title="PDIFF_6:th.11">Th11</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="E21">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:22_2"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">xseq</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</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">yseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>  st <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">xseq</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">yseq</font> &amp; (  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">xseq</font> holds <br/><font color="Olive" title="b1">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> ) holds <br/> <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b1">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">xseq</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">yseq</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">xseq</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">xseq</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">yseq</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">xseq</font> holds <br/><font color="Maroon" title="c6">yseq</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> ) implies  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c6">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">xseq</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:22_2_1"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">xseq</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">xseq</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">yseq</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">xseq</font> holds <br/><font color="Maroon" title="c6">yseq</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c6">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">xseq</font></span>)</span></span><br/><span class="kw">set </span><font color="Maroon" title="c7">xseq0</font> = <font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">yseq0</font> = <font color="Maroon" title="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font>;<br/><a NAME="E2:22_2_1"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E4:22_2_1"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_2_1_1"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c9">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">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</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="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span><br/>

<a NAME="E2:22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c9">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="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T57" title="RELAT_1:th.57">RELAT_1:57</a></span>;<br/>
<a NAME="E3:22_2_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">xseq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="relat_1.html#T57" title="RELAT_1:th.57">RELAT_1:57</a></span>;<br/>
<a NAME="E4:22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c6">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/>
<a NAME="E5:22_2_1_1"/>
<font color="Maroon" title="c5">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:22_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E4:22_2_1_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</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="E5:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A10</font></span>: 
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E22">A3</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">v</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><span class="kw"> such that </span><br/><a NAME="E7:22_2_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">xseq</font></span>)</span> &amp;  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c9">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:22_2_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="pdiff_6.html#T15" target="_self" title="PDIFF_6:th.15">Th15</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E9:22_2_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
( <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">yseq</font></span>)</span> &amp;  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c6">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">yseq</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c10">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:22_2_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="pdiff_6.html#T15" target="_self" title="PDIFF_6:th.15">Th15</a></span>;<br/><a NAME="E10:22_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">xseq</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"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E23">A4</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="E11:22_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <font color="Maroon" title="c9">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E7:22_2_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E12:22_2_1"/>
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c6">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">xseq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E7:22_2_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="pdiff_6.html#D1" target="_self" title="PDIFF_6:def.1">Def1</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="E3:22_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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="E3:22"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="#E1:22"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E21">A2</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c4">xseq</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</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">yseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">yseq</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">xseq</font> holds <br/><font color="Olive" title="b1">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) holds <br/> <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b1">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">yseq</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</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">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">yseq</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">xseq</font> holds <br/><font color="Maroon" title="c5">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E4:22"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">yseq</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">xseq</font> holds <br/><font color="Maroon" title="c5">yseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E5:22"/>
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c5">yseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c4">xseq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E22">A13</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>
