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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> $1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<div><span class="lab"><font color="Green" title="E24">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">fp</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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="c3">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/><span class="kw">assume </span><a NAME="E1:39_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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="c3">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">sp</font> =  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c2">fp</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:39_1_1_1"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sp</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T126" title="RVSUM_1:th.126">RVSUM_1:126</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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="c3">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:39_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T124" title="RVSUM_1:th.124">RVSUM_1:124</a></span>;<br/>
<a NAME="E3:39_1_1"/>
for <font color="Olive" title="b1">fp</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</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">fp</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S2" title="FINSEQ_2:sch.2">FINSEQ_2:sch 2</a>(<a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E1:39_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:39_1_1"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c1">fp</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
