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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="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:93_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">j</font> = <font color="Maroon" title="c3">i</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</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="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></i>;<br/><b>set </b><font color="Maroon" title="c5">lhs</font> =  <a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c6">rhs</font> =  <a href="nat_3.html#K6" title="NAT_3:func.6">max</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">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">m</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font>;<br/><b>set </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font>;<br/><b>set </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span>;<br/><a NAME="E3:93_1"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c4">j</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="E4:93_1"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</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="ref" href="int_2.html#T4" title="INT_2:th.4">INT_2:4</a></i>;<br/><a NAME="E5:93_1"/><i><font color="Green" title="E50">A3</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></i>;<br/><a NAME="E6:93_1"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:93_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="nat_3.html#E4:93_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/><a NAME="E7:93_1"/><i><font color="Green" title="E52">A5</font></i>: 
not <font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span></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="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:93_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="nat_3.html#E4:93_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/><a NAME="E8:93_1"/><i><font color="Green" title="E53">A6</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></i>;<br/><a NAME="E9:93_1"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <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="nat_3.html#E3:93_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/><a NAME="E10:93_1"/><i><font color="Green" title="E55">A8</font></i>: 
not <font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">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="c2">n</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:93_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/><a NAME="E11:93_1"/><i><font color="Green" title="E56">A9</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></i>;<br/><a NAME="E12:93_1"/><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:93_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/><a NAME="E13:93_1"/><i><font color="Green" title="E58">A11</font></i>: 
not <font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</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="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:93_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/><a NAME="E14:93_1"/><i><font color="Green" title="E59">A12</font></i>: 
<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> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">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="E15:93_1"/><i><font color="Green" title="E60">A13</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">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="E16:93_1"/><i><font color="Green" title="E61">A14</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:93_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="nat_3.html#E14:93_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/><a NAME="E17:93_1"/><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E9:93_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="nat_3.html#E15:93_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/><b>thus </b><a NAME="E18:93_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c2">n</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="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_1_1_1"/>
( <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</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="c1">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</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="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</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="c1">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1"><b>suppose </b></a><a NAME="E1:93_1_1_1_1"/><i><font color="Green" title="E63">A16</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</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="c1">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 ;<br/><div class="add"><a NAME="E2:93_1_1_1_1"/><b>then </b><i><font color="Green" title="E64">A17</font></i>: 
<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="c2">n</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="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E11:93_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="nat_3.html#D4" target="_self" title="NAT_3:def.4">Def4</a></i>;<br/><a NAME="E3:93_1_1_1_1"/><i><font color="Green" title="E65">A18</font></i>: 
<font color="Maroon" title="c4">j</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="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E7:93_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="nat_3.html#E16:93_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self" title="NAT_3:th.4">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_1_1"/>
<font color="Maroon" title="c4">j</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="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span>
 ;<br/><a NAME="E2:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</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="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:93_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self" title="NAT_3:th.4">Th4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:93_1_1_1_1_1"/><i><font color="Green" title="E66">A19</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</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/><i><font color="Green" title="E67">A20</font></i>: <font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="nat_3.html#E4:93_1_1_1_1_1"><i><font color="Green" title="E66">A19</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="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>

;<br/><a NAME="E6:93_1_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E4:93_1_1_1_1_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="ref" href="int_2.html#T4" title="INT_2:th.4">INT_2:4</a></i>;<br/><a NAME="E7:93_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E68">A21</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">k</font>
 ;<br/><a NAME="E8:93_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E69">A22</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:93_1_1_1_1_1"/><i><font color="Green" title="E70">A23</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E9:93_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:93_1_1_1_1_1"/><i><font color="Green" title="E71">A24</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E15:93_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:93_1_1_1_1_1"/><i><font color="Green" title="E72">A25</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">p</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:93_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="nat_3.html#E11:93_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="nat_3.html#E1:93_1_1_1_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">p</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">u</font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_1_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="nat_3.html#E10:93_1_1_1_1_1"><i><font color="Green" title="E70">A23</font></i></a>, <a class="txt" href="nat_3.html#E12:93_1_1_1_1_1"><i><font color="Green" title="E71">A24</font></i></a>, <a class="txt" href="nat_3.html#E14:93_1_1_1_1_1"><i><font color="Green" title="E72">A25</font></i></a>, <a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
;<br/><a NAME="E16:93_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E17:93_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E73">A26</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_1_1_3"/>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:93_1_1_1_1_1_3"/><i><font color="Green" title="E74">A27</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</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="c2">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</font>
<b>by </b><i><a class="txt" href="nat_3.html#E10:93_1_1_1_1_1"><i><font color="Green" title="E70">A23</font></i></a>, <a class="txt" href="nat_3.html#E3:93_1_1_1_1_1_3"><i><font color="Green" title="E74">A27</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:93_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E10:93_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E19:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E17:93_1_1_1_1_1"><i><font color="Green" title="E73">A26</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E21:93_1_1_1_1_1"/><i><font color="Green" title="E74">A28</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</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="E22:93_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">w</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_1_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="nat_3.html#E12:93_1_1_1_1_1"><i><font color="Green" title="E71">A24</font></i></a>, <a class="txt" href="nat_3.html#E21:93_1_1_1_1_1"><i><font color="Green" title="E74">A28</font></i></a></i>;<br/><a NAME="E23:93_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E24:93_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E75">A29</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</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>consider </b><font color="Maroon" title="c15">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E26:93_1_1_1_1_1"/><i><font color="Green" title="E76">A30</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">t</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:93_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E28:93_1_1_1_1_1"/><i><font color="Green" title="E77">A31</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E14:93_1"><i><font color="Green" title="E59">A12</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="E29:93_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_1_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="nat_3.html#E26:93_1_1_1_1_1"><i><font color="Green" title="E76">A30</font></i></a>, <a class="txt" href="nat_3.html#E28:93_1_1_1_1_1"><i><font color="Green" title="E77">A31</font></i></a></i>;<br/><a NAME="E30:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">u</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E31:93_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E78">A32</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c15">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_1_1_4"/>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c15">t</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:93_1_1_1_1_1_4"/><i><font color="Green" title="E79">A33</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</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="c1">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</font>
<b>by </b><i><a class="txt" href="nat_3.html#E26:93_1_1_1_1_1"><i><font color="Green" title="E76">A30</font></i></a>, <a class="txt" href="nat_3.html#E3:93_1_1_1_1_1_4"><i><font color="Green" title="E79">A33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:93_1_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E13:93_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E33:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c16">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E31:93_1_1_1_1_1"><i><font color="Green" title="E78">A32</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E35:93_1_1_1_1_1"/><i><font color="Green" title="E79">A34</font></i>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</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="E36:93_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">w</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_1_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="nat_3.html#E28:93_1_1_1_1_1"><i><font color="Green" title="E77">A31</font></i></a>, <a class="txt" href="nat_3.html#E35:93_1_1_1_1_1"><i><font color="Green" title="E79">A34</font></i></a></i>;<br/><a NAME="E37:93_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E38:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">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="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</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="E39:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">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="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E24:93_1_1_1_1_1"><i><font color="Green" title="E75">A29</font></i></a>, <a class="ref" href="nat_d.html#D4" title="NAT_D:def.4">NAT_D:def 4</a></i>;<br/><a NAME="E40:93_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E4:93_1_1_1_1_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="nat_3.html#E8:93_1_1_1_1_1"><i><font color="Green" title="E69">A22</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E41:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E7:93_1_1_1_1_1"><i><font color="Green" title="E68">A21</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E42:93_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E43:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E44:93_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:93_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c2">n</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="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="nat_3.html#E2:93_1_1_1_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="nat_3.html#E3:93_1_1_1_1"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_2"><b>suppose </b></a><a NAME="E1:93_1_1_1_2"/><i><font color="Green" title="E63">A35</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</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="c1">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 ;<br/><div class="add"><a NAME="E2:93_1_1_1_2"/><b>then </b><i><font color="Green" title="E64">A36</font></i>: 
<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="c2">n</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="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:93_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="nat_3.html#D4" target="_self" title="NAT_3:def.4">Def4</a></i>;<br/><a NAME="E3:93_1_1_1_2"/><i><font color="Green" title="E65">A37</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E7:93_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="nat_3.html#E17:93_1"><i><font color="Green" title="E62">A15</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self" title="NAT_3:th.4">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_2_1"/>
<font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span>
 ;<br/><a NAME="E2:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">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="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:93_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self" title="NAT_3:th.4">Th4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:93_1_1_1_2_1"/><i><font color="Green" title="E66">A38</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</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/><i><font color="Green" title="E67">A39</font></i>: <font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="nat_3.html#E4:93_1_1_1_2_1"><i><font color="Green" title="E66">A38</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="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>

;<br/><a NAME="E6:93_1_1_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E4:93_1_1_1_2_1"><i><font color="Green" title="E66">A38</font></i></a>, <a class="ref" href="int_2.html#T4" title="INT_2:th.4">INT_2:4</a></i>;<br/><a NAME="E7:93_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E68">A40</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">k</font>
 ;<br/><a NAME="E8:93_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E69">A41</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:93_1_1_1_2_1"/><i><font color="Green" title="E70">A42</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:93_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:93_1_1_1_2_1"/><i><font color="Green" title="E71">A43</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E14:93_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:93_1_1_1_2_1"/><i><font color="Green" title="E72">A44</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">p</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:93_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="nat_3.html#E11:93_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="nat_3.html#E1:93_1_1_1_2"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">p</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">u</font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_2_1"><i><font color="Green" title="E67">A39</font></i></a>, <a class="txt" href="nat_3.html#E10:93_1_1_1_2_1"><i><font color="Green" title="E70">A42</font></i></a>, <a class="txt" href="nat_3.html#E12:93_1_1_1_2_1"><i><font color="Green" title="E71">A43</font></i></a>, <a class="txt" href="nat_3.html#E14:93_1_1_1_2_1"><i><font color="Green" title="E72">A44</font></i></a>, <a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
;<br/><a NAME="E16:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c11">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E17:93_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E73">A45</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_2_1_3"/>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:93_1_1_1_2_1_3"/><i><font color="Green" title="E74">A46</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</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="c1">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</font>
<b>by </b><i><a class="txt" href="nat_3.html#E10:93_1_1_1_2_1"><i><font color="Green" title="E70">A42</font></i></a>, <a class="txt" href="nat_3.html#E3:93_1_1_1_2_1_3"><i><font color="Green" title="E74">A46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:93_1_1_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E13:93_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E19:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E17:93_1_1_1_2_1"><i><font color="Green" title="E73">A45</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E21:93_1_1_1_2_1"/><i><font color="Green" title="E74">A47</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">w</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="E22:93_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">w</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_2_1"><i><font color="Green" title="E67">A39</font></i></a>, <a class="txt" href="nat_3.html#E12:93_1_1_1_2_1"><i><font color="Green" title="E71">A43</font></i></a>, <a class="txt" href="nat_3.html#E21:93_1_1_1_2_1"><i><font color="Green" title="E74">A47</font></i></a></i>;<br/><a NAME="E23:93_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E24:93_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E75">A48</font></i>: 
<font color="Maroon" title="c1">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="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</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>consider </b><font color="Maroon" title="c15">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E26:93_1_1_1_2_1"/><i><font color="Green" title="E76">A49</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">t</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E9:93_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E28:93_1_1_1_2_1"/><i><font color="Green" title="E77">A50</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E15:93_1"><i><font color="Green" title="E60">A13</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="E29:93_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_2_1"><i><font color="Green" title="E67">A39</font></i></a>, <a class="txt" href="nat_3.html#E26:93_1_1_1_2_1"><i><font color="Green" title="E76">A49</font></i></a>, <a class="txt" href="nat_3.html#E28:93_1_1_1_2_1"><i><font color="Green" title="E77">A50</font></i></a></i>;<br/><a NAME="E30:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">u</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E31:93_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E78">A51</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c15">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_2_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_2_1_4"/>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c15">t</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:93_1_1_1_2_1_4"/><i><font color="Green" title="E79">A52</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</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="c2">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</font>
<b>by </b><i><a class="txt" href="nat_3.html#E26:93_1_1_1_2_1"><i><font color="Green" title="E76">A49</font></i></a>, <a class="txt" href="nat_3.html#E3:93_1_1_1_2_1_4"><i><font color="Green" title="E79">A52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:93_1_1_1_2_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E10:93_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E33:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c16">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E31:93_1_1_1_2_1"><i><font color="Green" title="E78">A51</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E35:93_1_1_1_2_1"/><i><font color="Green" title="E79">A53</font></i>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">w</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="E36:93_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">w</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1_1_1_2_1"><i><font color="Green" title="E67">A39</font></i></a>, <a class="txt" href="nat_3.html#E28:93_1_1_1_2_1"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="nat_3.html#E35:93_1_1_1_2_1"><i><font color="Green" title="E79">A53</font></i></a></i>;<br/><a NAME="E37:93_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E38:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</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="E39:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">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="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E24:93_1_1_1_2_1"><i><font color="Green" title="E75">A48</font></i></a>, <a class="ref" href="nat_d.html#D4" title="NAT_D:def.4">NAT_D:def 4</a></i>;<br/><a NAME="E40:93_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E4:93_1_1_1_2_1"><i><font color="Green" title="E66">A38</font></i></a>, <a class="txt" href="nat_3.html#E8:93_1_1_1_2_1"><i><font color="Green" title="E69">A41</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E41:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">j</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="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E7:93_1_1_1_2_1"><i><font color="Green" title="E68">A40</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E42:93_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E43:93_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E44:93_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:93_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c2">n</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="c1">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:93_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="nat_3.html#E2:93_1_1_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="nat_3.html#E3:93_1_1_1_2"><i><font color="Green" title="E65">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:93"/>
 <a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="nat_3.html#K6" title="NAT_3:func.6">max</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">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


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