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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">f</font> being   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span><br/>

<a NAME="E2:20"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
;<br/>
<a NAME="E3:20"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">fn</font> = <font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c3">n</font> as   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E5:20_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E6:20_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="euler_2.html#D2" target="_self" onmouseover="rs('euler_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E7:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E8:20_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c3">n</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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E9:20_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c6">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<a NAME="E3:20_1_1"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D16" onmouseover="rs('finseq_1/D16')" onmouseout="rh()">FINSEQ_1:def 16</a></span>;<br/>
<a NAME="E4:20_1_1"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E6:20_1_1"/>
<font color="Maroon" title="c6">i</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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A17</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:20_1_1"/>
<font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">fn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E3:20_1_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1_1"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:20_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="ref" href="finseq_1.html#D16" onmouseover="rs('finseq_1/D16')" onmouseout="rh()">FINSEQ_1:def 16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E10:20_1"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E11:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E12:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font> as   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E14:20_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E7:20_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="euler_2.html#D2" target="_self" onmouseover="rs('euler_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E15:20_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="finseq_1.html#T14" onmouseover="rs('finseq_1/T14')" onmouseout="rh()">FINSEQ_1:14</a></span>;<br/>
<a NAME="E16:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E12:20_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="ref" href="rfinseq.html#T7" onmouseover="rs('rfinseq/T7')" onmouseout="rh()">RFINSEQ:7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E24">A19</font></span>: <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T96" onmouseover="rs('rvsum_1/T96')" onmouseout="rh()">RVSUM_1:96</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E18">A7</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="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T73" onmouseover="rs('nat_d/T73')" onmouseout="rh()">NAT_D:73</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="rfinseq.html#T7" onmouseover="rs('rfinseq/T7')" onmouseout="rh()">RFINSEQ:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T96" onmouseover="rs('rvsum_1/T96')" onmouseout="rh()">RVSUM_1:96</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:20_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20_1"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E4:20"/><span class="lab"><font color="Green" title="E15">A20</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_2"/>
<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:20_2"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font></span><br/>

<a NAME="E3:20_2"/><span class="lab"><font color="Green" title="E17">A22</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="ref" href="euler_2.html#D2" target="_self" onmouseover="rs('euler_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E4:20_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:20_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E17">A22</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E5:20"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:20"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="euler_2.html#K3" title="EULER_2:func.3">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
