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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b2">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b3">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <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 &amp; <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b2">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b3">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) implies <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">n9</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>;<br/>
<a NAME="E1:30"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
for <font color="Olive" title="b1">k9</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st ( for <font color="Olive" title="b2">n9</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">n9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k9</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n9</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k9</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k9</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">n9</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">n9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k9</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n9</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k9</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
for <font color="Olive" title="b1">n9</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">n9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k9</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n9</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k9</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:30_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k9</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k9</font> &amp; <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) implies <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  or  not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  or  ex <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> or  not <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) or <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E2:30_1_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  or  ex <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> or  not <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) or <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E3:30_1_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  or  ex <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> or  not <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) or <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E4:30_1_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> or  not <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) or <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E5:30_1_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_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="E1:30_1_1_1"/>
( <font color="Maroon" title="c4">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c4">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c4">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:30_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:30_1_1_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:30_1_1_1_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a></span>;<br/><a NAME="E4:30_1_1_1_1"/><span class="kw">then </span>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T124" title="RVSUM_1:th.124">RVSUM_1:124</a></span>;<br/><span class="kw">hence </span><a NAME="E5:30_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="#E2:30_1_1_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c4">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">f9</font> = <font color="Maroon" title="c5">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></span>;<br/><a NAME="E3:30_1_1_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:30_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x1</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="txt" href="#E5:30_1_1_1_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a></span>;<br/><a NAME="E7:30_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_1_1_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></span>;<br/><a NAME="E8:30_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Del</a> <font color="Maroon" title="c6">f9</font>,<font color="Maroon" title="c8">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_1_1_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="nat_5.html#E29"><span class="lab"><font color="Green" title="E28">Lm8</font></span></a></span>;<br/><a NAME="E9:30_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T115" title="FINSEQ_3:th.115">FINSEQ_3:115</a></span>;<br/><a NAME="E10:30_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A15</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">n19</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>;<br/><a NAME="E11:30_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E12:30_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
 <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="nat_5.html#T6" target="_self" title="NAT_5:th.6">Th6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_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="E1:30_1_1_1_2_1"/>
( <font color="Maroon" title="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:30_1_1_1_2_1_1"/>
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">l</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <font color="Maroon" title="c10">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  ex <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  ex <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:30_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:30_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E6:30_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="#E5:30_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E9:30_1_1_1_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A19</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">thus </span><a NAME="E7:30_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E4:30_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E5:30_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="ref" href="nat_5.html#T8" target="_self" title="NAT_5:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">thus </span><a NAME="E8:30_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="#E5:30_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E6:30_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:30_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="#E1:30_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E8:30_1_1_1_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="txt" href="#E10:30_1_1_1_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E12:30_1_1_1_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E7:30_1_1_1_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E11:30_1_1_1_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E46">A23</font></span>: 
<font color="Maroon" title="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">f99</font> =  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>;<br/><a NAME="E2:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E47">A24</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_1_1_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E48">A25</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_1_1_2_1_2"/>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E6:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E50">A27</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="#E5:30_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E46">A23</font></span></a></span>;<br/><a NAME="E8:30_1_1_1_2_1_2"/>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="txt" href="#E6:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="ref" href="nat_5.html#T8" target="_self" title="NAT_5:th.8">Th8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E51">A28</font></span>: 
<font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">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="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E52">A29</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x2</font> = <font color="Maroon" title="c13">x2</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="txt" href="#E10:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A28</font></span></a></span>;<br/><a NAME="E13:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E53">A30</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Del</a> <font color="Maroon" title="c6">f9</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T115" title="FINSEQ_3:th.115">FINSEQ_3:115</a></span>;<br/><a NAME="E14:30_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A31</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_1_1_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E15:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E55">A32</font></span>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E56">A33</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E2:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E2:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E57">A34</font></span>: 
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E3:30_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E2:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:30_1_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A35</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:30_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E4:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:30_1_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A36</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">l9</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">l9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> &amp; <font color="Maroon" title="c15">l9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  ex <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">l9</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">l9</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<a NAME="E7:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E60">A37</font></span>: 
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E61">A38</font></span>: 
<font color="Maroon" title="c15">l9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">l9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  ex <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">l9</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">l9</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<a NAME="E9:30_1_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A39</font></span>: 
<font color="Maroon" title="c15">l9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Del</a> <font color="Maroon" title="c6">f9</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a></span>;<br/>
<a NAME="E10:30_1_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">l9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_1_1_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a></span>;<br/>
<a NAME="E11:30_1_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A40</font></span>: 
<font color="Maroon" title="c15">l9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:30_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c15">l9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">l9</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">l9</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k9</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E14:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E64">A41</font></span>: 
<font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E15:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E65">A42</font></span>: 
( <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">l9</font> )
 <span class="kw">and </span><br/><a NAME="E16:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E66">A43</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">l9</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">k9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="#E5:30_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E9:30_1_1_1_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E9:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">k9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> &amp; <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">l9</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">l9</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">k9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E17:30_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E67">A44</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E4:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E18:30_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">l9</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">l9</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">k9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E68">A45</font></span>: 
not <font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2_1_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="E1:30_1_1_1_2_1_2_1_1_1"/>
( <font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> or  not <font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A46</font></span>: 
<font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:30_1_1_1_2_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A47</font></span>: 
<font color="Maroon" title="c15">l9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E63">A40</font></span></a>, <a class="txt" href="#E7:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:30_1_1_1_2_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c15">l9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E11:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A29</font></span></a>, <a class="txt" href="#E16:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E66">A43</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E68">A45</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E69">A46</font></span></a>, <a class="ref" href="nat_5.html#T8" target="_self" title="NAT_5:th.8">Th8</a></span>;<br/><a NAME="E4:30_1_1_1_2_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">l9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E17:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E67">A44</font></span></a>, <a class="txt" href="#E6:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="txt" href="#E4:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="txt" href="#E2:30_1_1_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E70">A47</font></span></a>, <a class="ref" href="nat_5.html#T20" target="_self" title="NAT_5:th.20">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E5:30_1_1_1_2_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E5:30_1_1_1_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E13:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E8:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="ref" href="nat_5.html#T7" target="_self" title="NAT_5:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E69">A48</font></span>: 
not <font color="Maroon" title="c16">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:30_1_1_1_2_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E70">A49</font></span>: 
<font color="Maroon" title="c15">l9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E63">A40</font></span></a>, <a class="txt" href="#E7:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:30_1_1_1_2_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c15">l9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E14:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="txt" href="#E16:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E66">A43</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="ref" href="nat_5.html#T8" target="_self" title="NAT_5:th.8">Th8</a></span>;<br/><a NAME="E4:30_1_1_1_2_1_2_1_1_1_2"/><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="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E11:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A29</font></span></a>, <a class="txt" href="#E8:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E6:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="txt" href="#E4:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="txt" href="#E2:30_1_1_1_2_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E70">A49</font></span></a>, <a class="ref" href="nat_5.html#T20" target="_self" title="NAT_5:th.20">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E5:30_1_1_1_2_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E10:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="ref" href="nat_5.html#T6" target="_self" title="NAT_5:th.6">Th6</a>, <a class="ref" href="nat_5.html#T7" target="_self" title="NAT_5:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E19:30_1_1_1_2_1_2_1"/>
( <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c16">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">l9</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">l9</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">k9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</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="#E15:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E65">A42</font></span></a>, <a class="txt" href="#E16:30_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E66">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c15">n299</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span>;<br/><a NAME="E16:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E56">A50</font></span>: 
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E17:30_1_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Del</a> <font color="Maroon" title="c6">f9</font>,<font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></span>;<br/><a NAME="E18:30_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A51</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E7:30_1_1_1_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E11:30_1_1_1_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E16:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E56">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E19:30_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E58">A52</font></span>: 
 <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_1_1_1_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="nat_5.html#T6" target="_self" title="NAT_5:th.6">Th6</a></span>;<br/><a NAME="E20:30_1_1_1_2_1_2"/>
<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> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="txt" href="nat_5.html#E29"><span class="lab"><font color="Green" title="E28">Lm8</font></span></a></span>;<br/><a NAME="E21:30_1_1_1_2_1_2"/><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> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_1_1_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="txt" href="#E11:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A29</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> = 
<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> <span class="p4">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c14">x2</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T9" title="EULER_2:th.9">EULER_2:9</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E7:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E14:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E19:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E58">A52</font></span></a>, <a class="txt" href="#E15:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="txt" href="#E18:30_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E57">A51</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E2:30"/>
for <font color="Olive" title="b1">n9</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n9</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E31">A1</font></span></a>);<br/></span>
<a NAME="E3:30"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E4:30"/>
( <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">l</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">l</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) implies <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
