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<b>let </b><font color="Maroon" title="c1">n1</font>, <font color="Maroon" title="c2">n2</font>, <font color="Maroon" title="c3">n3</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> ;<br/>

<b>set </b><font color="Maroon" title="c4">d1</font> = <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font>;<br/>
<b>set </b><font color="Maroon" title="c5">d2</font> = <font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font>;<br/>
<b>set </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <font color="Maroon" title="c2">n2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">d</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <font color="Maroon" title="c2">n2</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">M</font> = <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <font color="Maroon" title="c2">n2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="ref" href="int_2.html#T4" title="INT_2:th.4">INT_2:4</a></i>;<br/>
<a NAME="E3:65"/><i><font color="Green" title="E46">A1</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n3</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n3</font> &amp; <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n2</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E4:65"/><b>then </b><i><font color="Green" title="E47">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n3</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D4" title="NAT_D:def.4">NAT_D:def 4</a></i>;<br/>
<a NAME="E5:65"/>
( <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">M</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">M</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D4" title="NAT_D:def.4">NAT_D:def 4</a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
( <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">M</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">M</font> )
 
<b>by </b><i><a class="txt" href="int_4.html#E3:65"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/>
<a NAME="E7:65"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">M</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D4" title="NAT_D:def.4">NAT_D:def 4</a></i>;<br/>
<a NAME="E8:65"/><b>then </b><i><font color="Green" title="E48">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">d</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E4:65"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E9:65"/><i><font color="Green" title="E49">B1</font></i>: 
for <font color="Olive" title="b1">p</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not <font color="Olive" title="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#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c7">d</font></span>)</span> or <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n1</font> or <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">p</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c9">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="c7">d</font></span>)</span> &amp; not <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n1</font> &amp; not <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n2</font> )
 ;<br/>

<a NAME="E2:65_1"/><b>then </b><i><font color="Green" title="E51">A6</font></i>: 
( not <font color="Maroon" title="c9">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n1</font> &amp; not <font color="Maroon" title="c9">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n2</font> )
 
<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="E3:65_1"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c9">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:65_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/>
<a NAME="E4:65_1"/><i><font color="Green" title="E53">A9</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">M</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E5:65_1"/>
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E54">A10</font></i>: <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">M</font> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font>
<b>by </b><i><a class="txt" href="int_4.html#E3:65_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="nat_3.html#D8" title="NAT_3:def.8">NAT_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nat_3.html#K6" title="NAT_3:func.6">max</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n2</font></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font>
<b>by </b><i><a class="ref" href="nat_3.html#T54" title="NAT_3:th.54">NAT_3:54</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1_2"/>
( <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n1</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> or <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n1</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_1"><b>suppose </b></a><a NAME="E1:65_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="c1">n1</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">M</font> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font>
<b>by </b><i><a class="txt" href="int_4.html#E1:65_1_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="nat_3.html#D4" title="NAT_3:def.4">NAT_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n2</font>
<b>by </b><i><a class="txt" href="int_4.html#E3:65_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="nat_3.html#D8" title="NAT_3:def.8">NAT_3:def 8</a></i>
;<br/><b>hence </b><a NAME="E3:65_1_1_2_1"/>
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">M</font>
 <b>by </b><i><a class="txt" href="int_4.html#E3:65_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="int_4.html#E2:65_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="moebius1.html#T9" title="MOEBIUS1:th.9">MOEBIUS1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_2"><b>suppose </b></a><a NAME="E1:65_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n1</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">M</font> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n1</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font>
<b>by </b><i><a class="txt" href="int_4.html#E1:65_1_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="nat_3.html#D4" title="NAT_3:def.4">NAT_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n1</font>
<b>by </b><i><a class="txt" href="int_4.html#E3:65_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="nat_3.html#D8" title="NAT_3:def.8">NAT_3:def 8</a></i>
;<br/><b>hence </b><a NAME="E3:65_1_1_2_2"/>
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">M</font>
 <b>by </b><i><a class="txt" href="int_4.html#E3:65_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="int_4.html#E2:65_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="moebius1.html#T9" title="MOEBIUS1:th.9">MOEBIUS1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:65_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_4.html#E4:65_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="int_4.html#E3:65_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="nat_3.html#T30" title="NAT_3:th.30">NAT_3:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:65"/><i><font color="Green" title="E50">B2</font></i>: 
for <font color="Olive" title="b1">p</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">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="c7">d</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">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="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">p</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:65_2"/><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c9">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="c7">d</font></span>)</span>
 ;<br/>

<a NAME="E2:65_2"/><i><font color="Green" title="E52">A13</font></i>: 
<font color="Maroon" title="c9">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:65_2"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/>
<a NAME="E3:65_2"/>
<font color="Maroon" title="c7">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n3</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E4:65_2"/><b>then </b><i><font color="Green" title="E53">A14</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n3</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E2:65_2"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="nat_3.html#T30" title="NAT_3:th.30">NAT_3:30</a></i>;<br/>
<a NAME="E5:65_2"/>
( <font color="Maroon" title="c9">p</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="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</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="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n3</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="E6:65_2"/><b>then </b>
<font color="Maroon" title="c9">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E4:65_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="radix_1.html#T7" title="RADIX_1:th.7">RADIX_1:7</a></i>;<br/>
<a NAME="E7:65_2"/><b>then </b><i><font color="Green" title="E54">A15</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:65_2"><i><font color="Green" title="E51">A11</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="E8:65_2"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E2:65_2"><i><font color="Green" title="E52">A13</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="E9:65_2"/><b>then </b>
<font color="Maroon" title="c9">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n3</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/>
<a NAME="E10:65_2"/><b>then </b><i><font color="Green" title="E55">A16</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n3</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E7:65_2"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/>
<a NAME="E11:65_2"/><i><font color="Green" title="E56">A17</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> &amp; <font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D4" title="NAT_D:def.4">NAT_D:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_2_1"/>
( <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n1</font> or <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n2</font> )
 </span><b>by </b><i><a class="txt" href="int_4.html#E1:65_2"><i><font color="Green" title="E51">A11</font></i></a>, <a class="txt" href="int_4.html#E9:65"><i><font color="Green" title="E49">B1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1_1"><b>suppose </b></a><a NAME="E1:65_2_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n1</font>
 ;<br/><div class="add"><a NAME="E2:65_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font>
 <b>by </b><i><a class="txt" href="int_4.html#E10:65_2"><i><font color="Green" title="E55">A16</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/><b>hence </b><a NAME="E3:65_2_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">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="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E11:65_2"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1_2"><b>suppose </b></a><a NAME="E1:65_2_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n2</font>
 ;<br/><div class="add"><a NAME="E2:65_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font>
 <b>by </b><i><a class="txt" href="int_4.html#E10:65_2"><i><font color="Green" title="E55">A16</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/><b>hence </b><a NAME="E3:65_2_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">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="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E11:65_2"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:65"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c9">g</font> =  <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font>;<br/>
<b>set </b><font color="Maroon" title="c10">K</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</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="c7">d</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c11">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E2:65_3"/><i><font color="Green" title="E51">A18</font></i>: 
(  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K20" title="RVSUM_1:func.20">Product</a> <font color="Maroon" title="c11">f</font> &amp; <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_3.html#D5" title="NAT_3:def.5">NAT_3:def 5</a></i>;<br/>
<a NAME="E3:65_3"/><i><font color="Green" title="E52">A19</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span></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#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" title="UPROOTS:th.5">UPROOTS:5</a></i>;<br/>
<a NAME="E4:65_3"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="txt" href="int_4.html#E3:65_3"><i><font color="Green" title="E52">A19</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">K</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</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="c7">d</font></span>)</span></span>)</span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E6:65_3"/><i><font color="Green" title="E53">D</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">K</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">z</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><br/>

<b>assume </b><a NAME="E1:65_3_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">K</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:65_3_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:65_3"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="seqm_3.html#D8" title="SEQM_3:def.8">SEQM_3:def 8</a></i>;<br/>
<a NAME="E8:65_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E6:65_3"><i><font color="Green" title="E53">D</font></i></a>, <a class="txt" href="int_4.html#E2:65_3"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c11">f</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E10:65_3"/><i><font color="Green" title="E54">B3</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">m</font>, <font color="Maroon" title="c15">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:65_3_2"/><i><font color="Green" title="E55">A21</font></i>: 
( <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c14">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">n</font> )
 ;<br/>

<a NAME="E2:65_3_2"/><b>then </b><i><font color="Green" title="E56">A22</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> &amp; <font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_4.html#E2:65_3"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E3:65_3_2"/><i><font color="Green" title="E57">A23</font></i>: 
( <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">K</font> &amp; <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">K</font> )
 
<b>by </b><i><a class="txt" href="int_4.html#E2:65_3"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="int_4.html#E1:65_3_2"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E4:65_3_2"/><b>then </b>
( <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">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="c7">d</font></span>)</span> &amp; <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</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="c7">d</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_4.html#E3:65_3"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:65_3_2"/><b>then </b><i><font color="Green" title="E58">A24</font></i>: 
( <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">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#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c7">d</font></span>)</span> &amp; <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</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="c7">d</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="E6:65_3_2"/><b>then </b><i><font color="Green" title="E59">A25</font></i>: 
( <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font> is <a href="int_2.html#V1" title="INT_2:attr.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="E7:65_3_2"/><i><font color="Green" title="E60">A26</font></i>: 
( <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="int_4.html#E5:65_3_2"><i><font color="Green" title="E58">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="E8:65_3_2"/>
<font color="Maroon" title="c12">K</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="uproots.html#T4" title="UPROOTS:th.4">UPROOTS:4</a></i>;<br/>
<a NAME="E9:65_3_2"/><b>then </b>
<font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E3:65_3_2"><i><font color="Green" title="E57">A23</font></i></a>, <a class="txt" href="int_4.html#E1:65_3_2"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E10:65_3_2"/><b>then </b>
<font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font>,<font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E6:65_3_2"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></i>;<br/>
<a NAME="E11:65_3_2"/><b>then </b><i><font color="Green" title="E61">A27</font></i>: 
<font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <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="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E7:65_3_2"><i><font color="Green" title="E60">A26</font></i></a>, <a class="ref" href="euler_2.html#T32" title="EULER_2:th.32">EULER_2:32</a></i>;<br/>
<a NAME="E12:65_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <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="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="int_4.html#E7:65_3_2"><i><font color="Green" title="E60">A26</font></i></a>, <a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E13:65_3_2"/><b>then </b><i><font color="Green" title="E62">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <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="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <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="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E11:65_3_2"><i><font color="Green" title="E61">A27</font></i></a>, <a class="txt" href="int_4.html#E7:65_3_2"><i><font color="Green" title="E60">A26</font></i></a>, <a class="ref" href="euler_2.html#T32" title="EULER_2:th.32">EULER_2:32</a></i>;<br/>
<a NAME="E14:65_3_2"/>
( <font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <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="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> &amp; <font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <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="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">d</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_4.html#E2:65_3_2"><i><font color="Green" title="E56">A22</font></i></a>, <a class="txt" href="int_4.html#E5:65_3_2"><i><font color="Green" title="E58">A24</font></i></a>, <a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E15:65_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a>, <a class="txt" href="int_4.html#E13:65_3_2"><i><font color="Green" title="E62">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:65_3"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> holds <br/><font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:65_3_3"/><i><font color="Green" title="E55">A29</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font>
 ;<br/>

<a NAME="E2:65_3_3"/><b>then </b>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">K</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E2:65_3"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E3:65_3_3"/><b>then </b>
<font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">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="c7">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E3:65_3"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:65_3_3"/><b>then </b>
<font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">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#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c7">d</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="E5:65_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">K</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E10:65"><i><font color="Green" title="E50">B2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:65_3_3"/>
<font color="Maroon" title="c13">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E2:65_3"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="int_4.html#E1:65_3_3"><i><font color="Green" title="E55">A29</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:65_3"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E2:65_3"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="int_4.html#E10:65_3"><i><font color="Green" title="E54">B3</font></i></a>, <a class="ref" href="int_4.html#T38" target="_self" title="INT_4:th.38">Th35</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:65"/><b>then </b>
<font color="Maroon" title="c7">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#T61" title="NAT_3:th.61">NAT_3:61</a></i>;<br/>
<b>hence </b><a NAME="E13:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n2</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <font color="Maroon" title="c2">n2</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">n3</font>
 <b>by </b><i><a class="txt" href="int_4.html#E8:65"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="nat_d.html#T5" title="NAT_D:th.5">NAT_D:5</a></i>;<br/>


</div>
