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

<b>let </b><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> ;<br/><b>let </b><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>;<br/>



<b>defpred </b><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> ] <b>means </b>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#K2" title="SEQ_1:func.2">.</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/>
<a NAME="E1:51"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E35">A2</font></i>: 
(  <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> 0 &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="c3">f</font> )
 ;<br/>

<a NAME="E2:51_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/>
<b>then </b><i><font color="Green" title="E36">A3</font></i>: <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#K2" title="SEQ_1:func.2">.</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> 0</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#T21" target="_self" title="RFUNCT_3:th.21">Th21</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E4:51_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:51_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:51_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:51"/><i><font color="Green" title="E35">A4</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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> ;<br/>

<b>assume </b><a NAME="E1:51_2"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/>

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E2:51_2"/><i><font color="Green" title="E37">A6</font></i>: 
(  <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> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.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="Maroon" title="c4">f</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">fn</font> = <font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E3:51_2"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:51_2"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
 <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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E5:51_2"/><i><font color="Green" title="E39">A8</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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="c4">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="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="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="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>)</span> &amp;  <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:51_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:51_2"/><b>then </b><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>  <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E9:51_2"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">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="c6">G</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>;<br/>
<a NAME="E10:51_2"/><b>then </b><i><font color="Green" title="E41">A10</font></i>: 
 <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">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="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T23" target="_self" title="RFUNCT_3:th.23">Th23</a></i>;<br/>
<a NAME="E11:51_2"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="rfunct_3.html#E7:51_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="rfunct_3.html#D9" target="_self" title="RFUNCT_3:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_2_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1"><b>case </b></a><a NAME="E1:51_2_1_1_1"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:51_2_1_1_1"/><b>then </b><i><font color="Green" title="E44">A13</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">G</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E3:51_2_1_1_1"/><i><font color="Green" title="E45">A14</font></i>: 
(  <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> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="rfunct_3.html#E1:51_2_1_1_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E4:51_2_1_1_1"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
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="c4">f</font> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c2">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">m</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> ;<br/><b>assume </b><a NAME="E1:51_2_1_1_1_1"/>
<font color="Maroon" title="c7">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
 ;<br/><a NAME="E2:51_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E47">A16</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font> = 
<font color="Maroon" title="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:51_2_1_1_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="rfunct_3.html#E4:51_2_1_1_1"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></i>
<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="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2_1_1_1_1"><i><font color="Green" title="E47">A16</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E6:51_2_1_1_1"/><b>then </b><i><font color="Green" title="E47">A17</font></i>: 
<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="relset_1.html#R2" title="RELSET_1:pred.2">=</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="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:51_2_1_1_1"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/><b>thus </b><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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font> = 
<font color="Maroon" title="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2_1_1_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="txt" href="rfunct_3.html#E6:51_2_1_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2"><b>case </b></a><a NAME="E1:51_2_1_1_2"/><i><font color="Green" title="E43">A18</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:51_2_1_1_2"/><b>then </b><i><font color="Green" title="E44">A19</font></i>: 
<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 href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="rfunct_3.html#T29" target="_self" title="RFUNCT_3:th.29">Th29</a></i>;<br/><a NAME="E3:51_2_1_1_2"/><b>then </b>
( <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="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">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> &amp; <font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:51_2"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rfunct_3.html#E4:51_2"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="rfunct_3.html#E1:51_2_1_1_2"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="rfunct_3.html#T31" target="_self" title="RFUNCT_3:th.31">Th31</a></i>;<br/><a NAME="E4:51_2_1_1_2"/><b>then </b>
<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#K4" title="RELSET_1:func.4">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="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E11:51_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:51_2_1_1_2"/><b>then </b><i><font color="Green" title="E45">A20</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <font color="Maroon" title="c6">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/><a NAME="E6:51_2_1_1_2"/><i><font color="Green" title="E46">A21</font></i>: 
<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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E5:51_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E7:51_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="rfunct_3.html#D8" target="_self" title="RFUNCT_3:def.8">Def8</a></i>;<br/><b>thus </b><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#K2" title="SEQ_1:func.2">.</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="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E10:51_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rfunct_3.html#E5:51_2_1_1_2"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>
<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="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">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#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:51_2"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rfunct_3.html#E4:51_2"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="rfunct_3.html#E2:51_2_1_1_2"><i><font color="Green" title="E44">A19</font></i></a></i>
<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="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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T32" target="_self" title="RFUNCT_3:th.32">Th32</a></i>
<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="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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">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="c6">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:51_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="rfunct_3.html#E5:51_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E6:51_2_1_1_2"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:51_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#K2" title="SEQ_1:func.2">.</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>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:51"/><i><font color="Green" title="E36">A22</font></i>: 
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="rfunct_3.html#E1:51"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="rfunct_3.html#E2:51"><i><font color="Green" title="E35">A4</font></i></a>);<br/></i>
<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E4:51"/><i><font color="Green" title="E37">A23</font></i>: 
<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>
 ;<br/>

<a NAME="E5:51"/>
 <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/>
<b>hence </b><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#K2" title="SEQ_1:func.2">.</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>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:51"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="rfunct_3.html#E4:51"><i><font color="Green" title="E37">A23</font></i></a></i>;<br/>


</div>
