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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font> ) implies <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E19">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b>for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Olive" title="b1">k</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> $1 &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Olive" title="b2">l</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> $1 &amp; ( for <font color="Olive" title="b3">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b2">l</font> ) holds <br/><font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">l</font>;<br/>
<a NAME="E2:25"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">l</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c3">k</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">l</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">l</font> ) implies <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">l</font> )</span><br/>

<b>assume </b><a NAME="E1:25_1"/>
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c3">k</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">l</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">l</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">l</font></span><br/>

<a NAME="E2:25_1"/><b>then </b>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E3:25_1"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">l</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><b> such that </b><br/><a NAME="E3:25_1_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><b>thus </b><a NAME="E4:25_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">l</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="moebius1.html#E3:25_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="nat_3.html#T37" title="NAT_3:th.37">NAT_3:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">l</font></span><br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">l</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:25"/><i><font color="Green" title="E21">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/>

<b>let </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> ) implies <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:25_2"/><i><font color="Green" title="E23">A7</font></i>: 
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E3:25_2"/><i><font color="Green" title="E24">A8</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1"/>
( ( not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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="c3">k</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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="c3">k</font> ) or ( not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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="c3">k</font> &amp; not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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="c3">k</font> ) or  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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="c3">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1"/><i><font color="Green" title="E25">A9</font></i>: 
( not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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="c3">k</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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="c3">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font></span><br/><div class="add"><div><i><font color="Green" title="E26">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_1_1_1"/><i><font color="Green" title="E27">A11</font></i>: 
not <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:25_2_1_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font> )</span><br/>

<b>assume </b><a NAME="E1:25_2_1_1_1_1"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font></span><br/>

<a NAME="E2:25_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E29">A13</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/>
<a NAME="E3:25_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E30">A14</font></i>: 
<font color="Maroon" title="c6">x</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 
<b>by </b><i><a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">x</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2_1_1_1_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/>
<a NAME="E5:25_2_1_1_1_1"/><i><font color="Green" title="E31">A15</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_1_1_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:25_2_1_1_1_1"/>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:25_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_1_1_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:25_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:25_2_1_1_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_1_1_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E8:25_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E9:25_2_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_1_1_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:25_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25_2_1_1"><i><font color="Green" title="E25">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>set </b><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font>;<br/><b>set </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font></span>)</span>;<br/><a NAME="E4:25_2_1_1"/><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2_1_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/><a NAME="E5:25_2_1_1"/><b>then </b><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c6">r</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 <b>by </b><i><a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/><a NAME="E6:25_2_1_1"/><b>then </b><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E7:25_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D7" title="NAT_3:def.7">NAT_3:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:25_2_1_1"/><i><font color="Green" title="E30">A19</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E10:25_2_1_1"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon" title="c9">t</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E9:25_2_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font></span>)</span>, <font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c9">t</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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>  <b>by </b><i><a class="txt" href="moebius1.html#E9:25_2_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E12:25_2_1_1"/><i><font color="Green" title="E32">A21</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c11">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/><a NAME="E13:25_2_1_1"/><i><font color="Green" title="E33">A22</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</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="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font> )</span><br/>

<b>assume </b><a NAME="E1:25_2_1_1_2"/><i><font color="Green" title="E34">A23</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font></span><br/>

<a NAME="E2:25_2_1_1_2"/><b>then </b><i><font color="Green" title="E35">A24</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/>
<a NAME="E3:25_2_1_1_2"/><b>then </b><i><font color="Green" title="E36">A25</font></i>: 
<font color="Maroon" title="c12">x</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 
<b>by </b><i><a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c12">x</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2_1_1_2"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/>
<a NAME="E5:25_2_1_1_2"/><i><font color="Green" title="E37">A26</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_1_2"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:25_2_1_1_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E9:25_2_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="nat_3.html#T45" title="NAT_3:th.45">NAT_3:45</a></i>;<br/>
<a NAME="E7:25_2_1_1_2"/><b>then </b>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E12:25_2_1_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/>
<a NAME="E8:25_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:25_2_1_1_2"><i><font color="Green" title="E34">A23</font></i></a></i>;<br/>
<a NAME="E9:25_2_1_1_2"/><b>then </b><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:25_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_1_1_2_1"/><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:25_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="nat_3.html#D8" title="NAT_3:def.8">NAT_3:def 8</a></i>;<br/><a NAME="E3:25_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2_1_1_2"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_1_2_1"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:25_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E5:25_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">g</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:25_2_1_1_2_1"/><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">g</font> = <font color="Maroon" title="c15">g</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:25_2_1_1_2_1"/>
<font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E6:25_2_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="nat_3.html#D7" title="NAT_3:def.7">NAT_3:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:25_2_1_1_2_1"/><i><font color="Green" title="E41">A30</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">u</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><font color="Maroon" title="c5">n</font> = 
<font color="Maroon" title="c10">s</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">u</font></span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E9:25_2_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="moebius1.html#E7:25_2_1_1_2_1"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="moebius1.html#E11:25_2_1_1_2_1"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">u</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">u</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><a NAME="E13:25_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>hence </b><a NAME="E14:25_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E6:25_2_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="nat_3.html#D7" title="NAT_3:def.7">NAT_3:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:25_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E9:25_2_1_1_2"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:25_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E13:25_2_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_1_2"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E14:25_2_1_1"/><i><font color="Green" title="E34">A31</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2_1_1_3"/>
<font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:25_2_1_1_3"/><b>then </b>
not <font color="Maroon" title="c6">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E6:25_2_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></i>;<br/>
<b>hence </b><a NAME="E3:25_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E4:25_2_1_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="nat_3.html#T36" title="NAT_3:th.36">NAT_3:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E15:25_2_1_1"/><i><font color="Green" title="E35">A32</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c10">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/><a NAME="E16:25_2_1_1"/><b>then </b><i><font color="Green" title="E36">A33</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="moebius1.html#E14:25_2_1_1"><i><font color="Green" title="E34">A31</font></i></a>, <a class="ref" href="nat_3.html#T42" title="NAT_3:th.42">NAT_3:42</a></i>;<br/><div><i><font color="Green" title="E37">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_1_1_4"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:25_2_1_1_4"/><i><font color="Green" title="E38">A35</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span>
 <b>and </b><br/><a NAME="E4:25_2_1_1_4"/><i><font color="Green" title="E39">A36</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:25_2_1_1_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E16:25_2_1_1"><i><font color="Green" title="E36">A33</font></i></a>, <a class="txt" href="moebius1.html#E3:25_2_1_1_4"><i><font color="Green" title="E38">A35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:25_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E13:25_2_1_1"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="moebius1.html#E4:25_2_1_1_4"><i><font color="Green" title="E39">A36</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:25_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E18:25_2_1_1"/>
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c5">n</font></span>)</span> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c11">t</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/><a NAME="E19:25_2_1_1"/><b>then </b><i><font color="Green" title="E38">A37</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E9:25_2_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="moebius1.html#E15:25_2_1_1"><i><font color="Green" title="E35">A32</font></i></a>, <a class="ref" href="nat_3.html#T46" title="NAT_3:th.46">NAT_3:46</a></i>;<br/><a NAME="E20:25_2_1_1"/><i><font color="Green" title="E39">A38</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span><br/>

<a NAME="E1:25_2_1_1_5"/><i><font color="Green" title="E40">A39</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E9:25_2_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="nat_3.html#T28" title="NAT_3:th.28">NAT_3:28</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1_1_5_1"/>
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span> or  not <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1_5_1_1"/><i><font color="Green" title="E41">A40</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1_1_5_1_1_1"/>
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span> or <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="moebius1.html#E19:25_2_1_1"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_1_5_1_1"><i><font color="Green" title="E41">A40</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1_5_1_1_1_1"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span><br/><div class="add"><a NAME="E2:25_2_1_1_5_1_1_1_1"/><b>then </b><i><font color="Green" title="E42">A41</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E16:25_2_1_1"><i><font color="Green" title="E36">A33</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1_1_5_1_1_1_1_1"/>
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</font></span>)</span> or  not <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1_5_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span><br/><div class="add"><a NAME="E2:25_2_1_1_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25_2_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:25_2_1_1_5_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2_1_1_5_1_1_1_1"><i><font color="Green" title="E42">A41</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_1_5_1_1_1_1_1_2"/>
not <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span><br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_1_5_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T15" target="_self" title="MOEBIUS1:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_1_5_1_1_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span><br/><div class="add"><a NAME="E2:25_2_1_1_5_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c10">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E17:25_2_1_1"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E3:25_2_1_1_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="moebius1.html#T15" target="_self" title="MOEBIUS1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E4:25_2_1_1_5_1_1_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_1_5"><i><font color="Green" title="E40">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_1_5_1_2"/>
not <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font></span><br/><div class="add"><a NAME="E2:25_2_1_1_5_1_2"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="moebius1.html#T15" target="_self" title="MOEBIUS1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:25_2_1_1_5_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2"><i><font color="Green" title="E24">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E21:25_2_1_1"/><i><font color="Green" title="E40">A42</font></i>: 
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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="c3">k</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</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="c3">k</font> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">t</font> ) implies <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t</font> )
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25_2"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/><a NAME="E22:25_2_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c11">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E20:25_2_1_1"><i><font color="Green" title="E39">A38</font></i></a>, <a class="ref" href="moebius1.html#T17" target="_self" title="MOEBIUS1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E23:25_2_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E10:25_2_1_1"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="moebius1.html#E13:25_2_1_1"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="moebius1.html#E20:25_2_1_1"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="moebius1.html#E21:25_2_1_1"><i><font color="Green" title="E40">A42</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_2"/><i><font color="Green" title="E25">A43</font></i>: 
( not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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="c3">k</font> &amp; not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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="c3">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font></span><br/><div class="add"><b>consider </b><font color="Maroon" title="c6">m1</font> being  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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> , <font color="Maroon" title="c7">e1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:25_2_1_2"/><i><font color="Green" title="E26">A44</font></i>: 
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</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="c3">k</font> &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c7">e1</font></span>)</span> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds <br/> ( ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> implies <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> ) &amp; ( not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> implies <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> ) ) ) )
 <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="moebius1.html#T18" target="_self" title="MOEBIUS1:th.18">Th18</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">m2</font> being  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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> , <font color="Maroon" title="c9">e2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:25_2_1_2"/><i><font color="Green" title="E27">A45</font></i>: 
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</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="c3">k</font> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m2</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c9">e2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds <br/> ( ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> implies <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> ) &amp; ( not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> implies <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> ) ) ) )
 <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="moebius1.html#T18" target="_self" title="MOEBIUS1:th.18">Th18</a></i>;<br/><a NAME="E6:25_2_1_2"/><i><font color="Green" title="E28">A46</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_2"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="moebius1.html#T16" target="_self" title="MOEBIUS1:th.16">Th16</a></i>;<br/><a NAME="E7:25_2_1_2"/>
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> holds  <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1_2_1_1"/>
( ( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> ) or ( not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> ) or ( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> ) or ( not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_2_1_1_1"/><i><font color="Green" title="E29">A47</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font></span><br/><div class="add"><a NAME="E2:25_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E30">A48</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_2"><i><font color="Green" title="E26">A44</font></i></a></i>;<br/><a NAME="E3:25_2_1_2_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_2"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_2_1_1_1"><i><font color="Green" title="E29">A47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:25_2_1_2_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="moebius1.html#E2:25_2_1_2_1_1_1"><i><font color="Green" title="E30">A48</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_2_1_1_2"/>
( not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_2_1_1_2"/>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T15" target="_self" title="MOEBIUS1:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:25_2_1_2_1_1_3"/><i><font color="Green" title="E29">A49</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font></span><br/><div class="add"><a NAME="E2:25_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c6">m1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/><a NAME="E3:25_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E30">A50</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m1</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T36" title="NAT_3:th.36">NAT_3:36</a></i>;<br/><a NAME="E4:25_2_1_2_1_1_3"/><i><font color="Green" title="E31">A51</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E5:25_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_2_1_1_3"><i><font color="Green" title="E30">A50</font></i></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></i>;<br/><a NAME="E6:25_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E32">A52</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E7:25_2_1_2_1_1_3"/>
<font color="Maroon" title="c6">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_2"><i><font color="Green" title="E26">A44</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E8:25_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T30" title="NAT_3:th.30">NAT_3:30</a></i>;<br/><a NAME="E9:25_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E33">A53</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="moebius1.html#E6:25_2_1_2_1_1_3"><i><font color="Green" title="E32">A52</font></i></a></i>;<br/><a NAME="E10:25_2_1_2_1_1_3"/>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25_2_1_2_1_1_3"><i><font color="Green" title="E29">A49</font></i></a>, <a class="ref" href="moebius1.html#T15" target="_self" title="MOEBIUS1:th.15">Th15</a></i>;<br/><a NAME="E11:25_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E34">A54</font></i>: 
not <font color="Maroon" title="c10">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">m2</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E4:25_2_1_2_1_1_3"><i><font color="Green" title="E31">A51</font></i></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></i>;<br/><a NAME="E12:25_2_1_2_1_1_3"/>
<font color="Maroon" title="c10">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E4:25_2_1_2_1_1_3"><i><font color="Green" title="E31">A51</font></i></a>, <a class="txt" href="moebius1.html#E9:25_2_1_2_1_1_3"><i><font color="Green" title="E33">A53</font></i></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></i>;<br/><a NAME="E13:25_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c9">e2</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_2"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="moebius1.html#E11:25_2_1_2_1_1_3"><i><font color="Green" title="E34">A54</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><a NAME="E14:25_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_3.html#T5" title="NAT_3:th.5">NAT_3:5</a></i>;<br/><a NAME="E15:25_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E35">A55</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E6:25_2_1_2"><i><font color="Green" title="E28">A46</font></i></a>, <a class="txt" href="moebius1.html#E4:25_2_1_2_1_1_3"><i><font color="Green" title="E31">A51</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E16:25_2_1_2_1_1_3"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E17:25_2_1_2_1_1_3"/>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_2"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_2_1_1_3"><i><font color="Green" title="E29">A49</font></i></a>, <a class="txt" href="moebius1.html#E15:25_2_1_2_1_1_3"><i><font color="Green" title="E35">A55</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:25_2_1_2_1_1_4"/>
( not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span> &amp; not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_2_1_1_4"/>
<font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T15" target="_self" title="MOEBIUS1:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:25_2_1_2"/><b>then </b><i><font color="Green" title="E29">A56</font></i>: 
<font color="Maroon" title="c6">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">m2</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="moebius1.html#E3:25_2_1_2"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="moebius1.html#E5:25_2_1_2"><i><font color="Green" title="E27">A45</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> <b>by </b><i><a class="txt" href="moebius1.html#E2:25_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_2"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="moebius1.html#T16" target="_self" title="MOEBIUS1:th.16">Th16</a></i>;<br/><a NAME="E10:25_2_1_2"/><i><font color="Green" title="E30">A57</font></i>: 
<font color="Maroon" title="c10">w</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E11:25_2_1_2"/><b>then </b><i><font color="Green" title="E31">A58</font></i>: 
<font color="Maroon" title="c10">w</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">e1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e1</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T25" title="NAT_3:th.25">NAT_3:25</a></i>;<br/><a NAME="E12:25_2_1_2"/><i><font color="Green" title="E32">A59</font></i>: 
<font color="Maroon" title="c10">w</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E10:25_2_1_2"><i><font color="Green" title="E30">A57</font></i></a>, <a class="ref" href="nat_3.html#T25" title="NAT_3:th.25">NAT_3:25</a></i>;<br/><a NAME="E13:25_2_1_2"/><i><font color="Green" title="E33">A60</font></i>: 
not <font color="Maroon" title="c10">w</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2_1_2_2"/>
<font color="Maroon" title="c10">w</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:25_2_1_2_2"/><b>then </b>
<font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c6">m1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#T37" title="NAT_3:th.37">NAT_3:37</a></i>;<br/>
<a NAME="E3:25_2_1_2_2"/><b>then </b>
<font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c6">m1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/>
<a NAME="E4:25_2_1_2_2"/><b>then </b>
<font color="Maroon" title="c10">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_2"><i><font color="Green" title="E26">A44</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:25_2_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><i><font color="Green" title="E34">A61</font></i>: <font color="Maroon" title="c10">w</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">m1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c7">e1</font>
<b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_2"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="moebius1.html#E11:25_2_1_2"><i><font color="Green" title="E31">A58</font></i></a>, <a class="ref" href="nat_3.html#T28" title="NAT_3:th.28">NAT_3:28</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c7">e1</font>
<b>by </b><i><a class="txt" href="moebius1.html#E10:25_2_1_2"><i><font color="Green" title="E30">A57</font></i></a>, <a class="txt" href="moebius1.html#E13:25_2_1_2"><i><font color="Green" title="E33">A60</font></i></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></i>
<br/>.= 
<font color="Maroon" title="c7">e1</font>

;<br/><a NAME="E15:25_2_1_2"/><i><font color="Green" title="E35">A62</font></i>: 
not <font color="Maroon" title="c10">w</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">m2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2_1_2_4"/>
<font color="Maroon" title="c10">w</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">m2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:25_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c8">m2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#T37" title="NAT_3:th.37">NAT_3:37</a></i>;<br/>
<a NAME="E3:25_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">m2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/>
<a NAME="E4:25_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c10">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_2"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:25_2_1_2_4"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><font color="Maroon" title="c10">w</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">m2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">e2</font>
<b>by </b><i><a class="txt" href="moebius1.html#E5:25_2_1_2"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="moebius1.html#E12:25_2_1_2"><i><font color="Green" title="E32">A59</font></i></a>, <a class="ref" href="nat_3.html#T28" title="NAT_3:th.28">NAT_3:28</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">e2</font>
<b>by </b><i><a class="txt" href="moebius1.html#E10:25_2_1_2"><i><font color="Green" title="E30">A57</font></i></a>, <a class="txt" href="moebius1.html#E15:25_2_1_2"><i><font color="Green" title="E35">A62</font></i></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></i>
<br/>.= 
<font color="Maroon" title="c9">e2</font>

;<br/><a NAME="E17:25_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c7">e1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c9">e2</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="moebius1.html#E14:25_2_1_2"><i><font color="Green" title="E34">A61</font></i></a>, <a class="ref" href="radix_1.html#T7" title="RADIX_1:th.7">RADIX_1:7</a></i>;<br/><b>hence </b><a NAME="E18:25_2_1_2"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2_1_2"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="moebius1.html#E5:25_2_1_2"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="moebius1.html#E8:25_2_1_2"><i><font color="Green" title="E29">A56</font></i></a>, <a class="ref" href="nat_3.html#T1" title="NAT_3:th.1">NAT_3:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_3"><b>suppose </b></a><a NAME="E1:25_2_1_3"/><i><font color="Green" title="E25">A63</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</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="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font></span><br/><div class="add"><a NAME="E2:25_2_1_3"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:25_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="moebius1.html#T17" target="_self" title="MOEBIUS1:th.17">Th17</a></i>;<br/><a NAME="E3:25_2_1_3"/><b>then </b>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c4">m</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="c3">k</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25_2_1_3"><i><font color="Green" title="E25">A63</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E4:25_2_1_3"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="moebius1.html#E3:25_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="moebius1.html#E1:25_2_1_3"><i><font color="Green" title="E25">A63</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:25"/><i><font color="Green" title="E22">A64</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="moebius1.html#E2:25"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="moebius1.html#E3:25"><i><font color="Green" title="E21">A5</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c3">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:25"/><i><font color="Green" title="E23">A65</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</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="c3">k</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T14" target="_self" title="MOEBIUS1:th.14">Th14</a></i>;<br/>
<a NAME="E7:25"/><i><font color="Green" title="E24">A66</font></i>: 
( <font color="Maroon" title="c1">m</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:25"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="moebius1.html#T17" target="_self" title="MOEBIUS1:th.17">Th17</a></i>;<br/>
<a NAME="E9:25"/><b>then </b>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">m</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="c3">k</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E6:25"><i><font color="Green" title="E23">A65</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E10:25"/>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="moebius1.html#E4:25"><i><font color="Green" title="E22">A64</font></i></a>, <a class="txt" href="moebius1.html#E6:25"><i><font color="Green" title="E23">A65</font></i></a>, <a class="txt" href="moebius1.html#E7:25"><i><font color="Green" title="E24">A66</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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