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

<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</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>   st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Olive" title="b1">f</font></span><br/>

<b>defpred </b><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> ] <b>means </b>for <font color="Olive" title="b1">a</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> $1 holds <br/><font color="Olive" title="b1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> $1;<br/>
<a NAME="E1:10"/><i><font color="Green" title="E7">A1</font></i>: 
<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> ]
 
;<br/>
<a NAME="E2:10"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">p</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> <br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">p</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</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">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>]</span><br/><b>let </b><font color="Maroon" title="c3">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>] )</span><br/>



<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>]</span><br/>

<b>set </b><font color="Maroon" title="c4">p1</font> = <font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>;<br/>
<b>let </b><font color="Maroon" title="c5">a</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> implies <font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:10_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span><br/>

<a NAME="E3:10_1"/><i><font color="Green" title="E11">A5</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E4:10_1"/><i><font color="Green" title="E12">A6</font></i>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T126" title="RVSUM_1:th.126">RVSUM_1:126</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> )
 </span><b>by </b><i><a class="txt" href="#E2:10_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="#E3:10_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="#E4:10_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E3:10_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:10_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:10_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:10"/>
for <font color="Olive" title="b1">p</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">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2" title="FINSEQ_2:sch.2">FINSEQ_2:sch 2</a>(<a class="txt" href="#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E8">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:10"/>
for <font color="Olive" title="b1">f</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>   st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Olive" title="b1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
