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<b>let </b><font color="Maroon" title="c1">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> ;<br/><b>let </b><font color="Maroon" title="c2">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> ;<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">b</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="Olive" title="b1">b</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> $1 holds <br/><font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> $1 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">b</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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><font color="Maroon" title="c2">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">Sum</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</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>( <font color="Maroon">S<sub>1</sub></font>[$1] implies <font color="Maroon">S<sub>2</sub></font>[$1] );<br/>
<a NAME="E1:52"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ]
 
<b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>;<br/>
<div><i><font color="Green" title="E38">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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> ;<br/><b>let </b><font color="Maroon" title="c4">d</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> ;<br/><b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font>]
 ;<br/><b>set </b><font color="Maroon" title="c5">fp1</font> = <font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 ;<br/><a NAME="E2:52_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">fp</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</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> ;<br/>

<b>assume </b><a NAME="E1:52_1_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c6">b</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="c3">fp</font>
 ;<br/>

<a NAME="E2:52_1_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">fp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E3:52_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:52_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:52_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="int_4.html#E1:52_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="int_4.html#E1:52_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="int_4.html#E2:52_1_1_1"><i><font color="Green" title="E43">A7</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:52_1_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">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <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"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:52_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</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="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E1:52_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/><a NAME="E5:52_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><a NAME="E6:52_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/><a NAME="E7:52_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c3">fp</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="int_4.html#E1:52_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="int_4.html#E2:52_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="nat_d.html#T8" title="NAT_D:th.8">NAT_D:8</a></i>;<br/><b>hence </b><a NAME="E8:52_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 <b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:52"/>
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>3</sub></font>[<font color="Olive" title="b1">fp</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="int_4.html#E1:52"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="int_4.html#E2:52"><i><font color="Green" title="E38">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:52"/>
( ( for <font color="Olive" title="b1">b</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="Olive" title="b1">b</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="c1">fp</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">b</font> ) implies <font color="Maroon" title="c2">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">Sum</a> <font color="Maroon" title="c1">fp</font> )
 ;<br/>


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