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<span class="kw">let </span><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> ; <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="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; (  for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b2">c</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="c1">fp</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; (  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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  for <font color="Olive" title="b1">fp1</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>   ex <font color="Olive" title="b2">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/> for <font color="Olive" title="b3">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="b3">b</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="c1">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">b</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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b2">c</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="c1">fp</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</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">  for <font color="Olive" title="b1">fp1</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>   ex <font color="Olive" title="b2">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/> for <font color="Olive" title="b3">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="b3">b</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="c1">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">b</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><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">fp2</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> , <font color="Maroon" title="c3">q</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> <span class="kw"> such that </span><br/><a NAME="E5:51"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c1">fp</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">fp2</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">q</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="#E1:51"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</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>  : <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,$1)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span> } </span> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>($1);<br/>
<a NAME="E6:51"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span> holds <br/> ex <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">a</font>,<font color="Olive" title="b2">n</font>]
 
<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">a</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="c4">a</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span> implies  ex <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="c4">a</font>,<font color="Olive" title="b1">n</font>] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">l</font> = <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">a</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">assume </span><a NAME="E2:51_1"/>
<font color="Maroon" title="c4">a</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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="c4">a</font>,<font color="Olive" title="b1">n</font>]</span><br/>

<a NAME="E3:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c4">a</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="c1">fp</font>
 
<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="E4:51_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">a</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="#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>, <a class="ref" href="int_4.html#T31" target="_self" title="INT_4:th.31">Th31</a></span>;<br/>
<a NAME="E5:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A7</font></span>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c5">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<a NAME="E6:51_1"/>
<font color="Maroon" title="c5">l</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:51_1"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">l</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">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="int_4.html#T16" target="_self" title="INT_4:th.16">Th16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">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">a</font>,<font color="Maroon" title="c7">n</font>]</span><br/>

<span class="kw">thus </span><a NAME="E10:51_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1"><span class="lab"><font color="Green" title="E41">A8</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><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">s</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> <span class="kw"> such that </span><br/><a NAME="E8:51"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">s</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="c1">fp</font></span>)</span> &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">a</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E38">A5</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c5">fp1</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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/> for <font color="Olive" title="b2">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="b2">b</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="c1">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">b</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><br/>

<span class="kw">set </span><font color="Maroon" title="c6">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,$1)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">s1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E10:51"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</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">s1</font> holds <br/><font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E11:51"/>
 for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</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">s1</font> holds <br/><font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c8">a</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">a</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">s1</font> implies <font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_2"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c8">a</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">s1</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" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:51_2"/>
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c9">a</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:51_2"/>
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">s1</font> = <font color="Maroon" title="c7">s1</font> as    <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">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">x</font> =  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font>;<br/>
<span class="kw">take </span>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</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">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</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><br/>

<a NAME="E13:51"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/>
<a NAME="E14:51"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">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">a</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="c1">fp</font> &amp; <font color="Olive" title="b2">b</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="c1">fp</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</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">  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="Maroon" title="c10">a</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="c1">fp</font> &amp; <font color="Olive" title="b1">b</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="c1">fp</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">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> ; <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="c10">a</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="c1">fp</font> &amp; <font color="Maroon" title="c11">b</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="c1">fp</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</font> implies <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51_3"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c10">a</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="c1">fp</font>
 <span class="kw">and </span><br/><a NAME="E2:51_3"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
( <font color="Maroon" title="c11">b</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="c1">fp</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</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" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span><br/>

<a NAME="E3:51_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<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="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:51_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">b</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">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">a</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E2:51_3"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="int_4.html#T35" target="_self" title="INT_4:th.35">Th35</a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E5:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">a</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
 
;<br/>
<a NAME="E6:51_3"/>
<font color="Maroon" title="c10">a</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="c8">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E1:51_3"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:51_3"/>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E5:51_3"><span class="lab"><font color="Green" title="E45">A16</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><span class="kw">end;</span></div>
<a NAME="E15:51"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Olive" title="b1">b</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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> ; <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="c10">b</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="c1">fp</font> implies <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_4"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<font color="Maroon" title="c10">b</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="c1">fp</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" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span><br/>

<a NAME="E2:51_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:51_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A19</font></span>: 
<font color="Maroon" title="c10">b</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="c8">s1</font>
 
<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="E4:51_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<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"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></span>;<br/>
<a NAME="E5:51_4"/>
 for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</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="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</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="c11">a</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="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> implies <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_4_1"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<font color="Maroon" title="c11">a</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="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span>
 ; <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="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span><br/>

<a NAME="E2:51_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A22</font></span>: 
( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:51_4_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T39" title="WSIERP_1:th.39">WSIERP_1:39</a></span>;<br/>
<a NAME="E4:51_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</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="c8">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_4_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/>
<a NAME="E5:51_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</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> <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="c2">fp2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></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_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<font color="Maroon" title="c11">a</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="c1">fp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_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_4_1_1"/>
( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">b</font> or <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_4_1_1_1"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">b</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" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span><br/><div class="add"><a NAME="E2:51_4_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_4"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_4_1_1_1"/>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E6:51_4_1"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="txt" href="#E1:51_4_1_1_1"><span class="lab"><font color="Green" title="E50">A24</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_4_1_1_2"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">b</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" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span><br/><div class="add"><a NAME="E2:51_4_1_1_2"/>
( <font color="Maroon" title="c11">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp; <font color="Maroon" title="c11">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4_1"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E3:51_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A26</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <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="c1">fp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:51_4_1_1_2"/>
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c11">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_4"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a>, <a class="ref" href="xreal_1.html#T39" title="XREAL_1:th.39">XREAL_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_4_1_1_2"/>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E1:51_4"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E3:51_4_1_1_2"><span class="lab"><font color="Green" title="E51">A26</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><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="E6:51_4"/>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T36" target="_self" title="INT_4:th.36">Th36</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="E16:51"/>
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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> ; <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="c10">b</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="c1">fp</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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><br/>

<span class="kw">assume </span><a NAME="E1:51_5"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
<font color="Maroon" title="c10">b</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="c1">fp</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"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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><br/>

<a NAME="E2:51_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A28</font></span>: 
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/>
<a NAME="E3:51_5"/><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E1:51_5"><span class="lab"><font color="Green" title="E44">A27</font></span></a></span>;<br/>
<a NAME="E4:51_5"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">l</font> =  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E2:51_5_1"/><span class="lab"><font color="Green" title="E48">A31</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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="#E3:51_5"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="txt" href="#E2:51_5"><span class="lab"><font color="Green" title="E45">A28</font></span></a>, <a class="ref" href="int_1.html#T62" title="INT_1:th.62">INT_1:62</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">l</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" title="NAT_D:th.66">NAT_D:66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_5_1"><span class="lab"><font color="Green" title="E48">A31</font></span></a>, <a class="ref" href="nat_d.html#T65" title="NAT_D:th.65">NAT_D:65</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:51_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</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="E5:51_5"/><span class="lab"><font color="Green" title="E48">A32</font></span>: 
( <font color="Maroon" title="c10">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E1:51_5"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:51_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<font color="Maroon" title="c10">b</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> <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="c2">fp2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E7:51_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</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="c8">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E50">A34</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>

;<br/>
<a NAME="E9:51_5"/>
<font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E6:51_5"><span class="lab"><font color="Green" title="E49">A33</font></span></a></span>;<br/>
<a NAME="E10:51_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A35</font></span>: 
 ex <font color="Olive" title="b1">ll</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 <br/>( <font color="Olive" title="b1">ll</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">ll</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<a NAME="E11:51_5"/>
<font color="Maroon" title="c10">b</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="c8">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E5:51_5"><span class="lab"><font color="Green" title="E48">A32</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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T20" title="WSIERP_1:th.20">WSIERP_1:20</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_5"><span class="lab"><font color="Green" title="E45">A28</font></span></a>, <a class="txt" href="#E8:51_5"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E10:51_5"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E4:51_5"><span class="lab"><font color="Green" title="E47">A30</font></span></a>, <a class="txt" href="int_4.html#E45"><span class="lab"><font color="Green" title="E32">Lm9</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:51_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:51"/>
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <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"> verum</span><br/>


</div>
