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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)  st <font color="Olive" title="b1">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b2">f</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Olive" title="b1">d</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</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">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)  st <font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span> for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c3">f</font> implies <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c3">f</font>
 ; <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"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span><br/>

<a NAME="E2:51"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c3">f</font>
 
;<br/>
<a NAME="E3:51"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 for <font color="Olive" title="b1">n</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#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">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>); <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">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c5">f</font> implies <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:51_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:51_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c5">f</font>
 ; <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"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">fn</font> = <font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font>;<br/>
<a NAME="E4:51_1"/><span class="lab"><font color="Green" title="E40">A7</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">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E38">A5</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="E5:51_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E38">A5</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="c7">G</font> = <font color="Maroon" title="c5">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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/>
<a NAME="E8:51_1"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span>)</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="E9:51_1"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <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="c7">G</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="rfinseq.html#T7" title="RFINSEQ:th.7">RFINSEQ:7</a></span>;<br/>
<a NAME="E10:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T20" target="_self" title="RFUNCT_3:th.20">Th23</a></span>;<br/>
<a NAME="E11:51_1"/><span class="lab"><font color="Green" title="E44">A11</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">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</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> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></span>;<br/>
<a NAME="E12:51_1"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E6:51_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="rfunct_3.html#D9" target="_self" title="RFUNCT_3:def.9">Def9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1"/>
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">n</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span><br/><div class="add"><a NAME="E2:51_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A14</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">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></span>;<br/><a NAME="E3:51_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A15</font></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">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><span class="lab"><font color="Green" title="E49">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">m</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="c8">m</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 implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:51_1_1_1_1_1"/>
<font color="Maroon" title="c8">m</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
 ; <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="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m</font></span><br/><a NAME="E2:51_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<font color="Maroon" title="c8">m</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><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m</font> = 
<font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
;<br/> <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="E5:51_1_1_1_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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="#E2:51_1_1_1_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:51_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1_1_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E4:51_1_1_1_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>;<br/><a NAME="E7:51_1_1_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">G</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> = 
<font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_1_1_1_1"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>
;<br/> <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span><br/><div class="add"><a NAME="E2:51_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E11:51_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E6:51_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></span>;<br/><a NAME="E3:51_1_1_1_2"/>
<font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_2"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="rfunct_3.html#T26" target="_self" title="RFUNCT_3:th.26">Th29</a></span>;<br/><a NAME="E4:51_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:51_1_1_1_2"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="rfunct_3.html#T28" target="_self" title="RFUNCT_3:th.28">Th31</a></span>;<br/><a NAME="E5:51_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:51_1"><span class="lab"><font color="Green" title="E45">A12</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="E6:51_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <font color="Maroon" title="c7">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:51_1_1_1_2"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="rfunct_3.html#T26" target="_self" title="RFUNCT_3:th.26">Th29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T29" target="_self" title="RFUNCT_3:th.29">Th32</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font></span>)</span></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="rvsum_1.html#T74" title="RVSUM_1:th.74">RVSUM_1:74</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E11:51_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E2:51_1_1_1_2"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="rfinseq.html#T7" title="RFINSEQ:th.7">RFINSEQ:7</a></span>
;<br/> <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="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:51_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</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="E4:51"/><span class="lab"><font color="Green" title="E37">A21</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="51_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>); <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">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c4">f</font> implies <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51_2"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</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="E2:51_2"/>
<font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c4">f</font>
 ; <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"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span></span><br/>

<a NAME="E3:51_2"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E39">A23</font></span>: <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T19" target="_self" title="RFUNCT_3:th.19">Th21</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/>
<a NAME="E5:51_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></span>;<br/>
<a NAME="E6:51_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E7:51_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</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="E5:51"/>
 for <font color="Olive" title="b1">n</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">n</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="#E4:51"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E36">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:51"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E35">A2</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>
