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<b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:94_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><b>set </b><font color="Maroon">lhs</font> =  <a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span>;<br/><b>set </b><font color="Maroon">rhs</font> =  <a href="nat_3.html#K6" target="_self">max</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span>;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font>;<br/><b>set </b><font color="Maroon">z</font> = <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span>;<br/><a NAME="E3:94_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E4:94_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="int_2.html#T4">INT_2:4</a></i>;<br/><a NAME="E5:94_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:94_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:94_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_3.html#E7:94_1"><i><font color="Green">E50</font></i></a>, </i>;<br/><a NAME="E7:94_1"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:94_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_3.html#E7:94_1"><i><font color="Green">E50</font></i></a>, </i>;<br/><a NAME="E8:94_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E9:94_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:94_1"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E10:94_1"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:94_1"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E11:94_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E12:94_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:94_1"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E13:94_1"/><i><font color="Green">E103</font></i>: 
not <font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:94_1"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E14:94_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/><a NAME="E15:94_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/><a NAME="E16:94_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><a NAME="E17:94_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, , <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><b>thus </b><a NAME="E18:94_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K6" target="_self">max</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1_1_1"/>
( <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> or <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1"><b>suppose </b></a><a NAME="E1:94_1_1_1_1"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K6" target="_self">max</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:94_1_1_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E10:94_1"><i><font color="Green">E53</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:94_1_1_1_1_1"/>
<font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><a NAME="E2:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E9:94_1"><i><font color="Green">E52</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:94_1_1_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E112</font></i>: <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span>

;<br/><a NAME="E6:94_1_1_1_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="int_2.html#T4">INT_2:4</a></i>;<br/><a NAME="E7:94_1_1_1_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E8:94_1_1_1_1_1"/><i><font color="Green">E114</font></i>: 
0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E11:94_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:94_1_1_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:94_1_1_1_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:94_1_1_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K3">|^</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K3">|^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a>, , <a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="nat_3.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_3.html#T13" target="_self">Th13</a></i>
;<br/><a NAME="E16:94_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K3">|^</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E17:94_1_1_1_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:94_1_1_1_1_1_3"/>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:94_1_1_1_1_1_3"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="nat_3.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:94_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E19:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:94_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E21:94_1_1_1_1_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E22:94_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_3.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E23:94_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E24:94_1_1_1_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:94_1_1_1_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E28:94_1_1_1_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E29:94_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_3.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E30:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E31:94_1_1_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:94_1_1_1_1_1_4"/>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:94_1_1_1_1_1_4"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="nat_3.html#T16" target="_self">Th16</a>, <a class="ref" href="nat_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:94_1_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E33:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E35:94_1_1_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E36:94_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="nat_3.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E37:94_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E38:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E39:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/><a NAME="E40:94_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/><a NAME="E41:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E11:94_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E42:94_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E43:94_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E44:94_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:94_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K6" target="_self">max</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:94_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_2"><b>suppose </b></a><a NAME="E1:94_1_1_1_2"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:94_1_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K6" target="_self">max</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E3:94_1_1_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E10:94_1"><i><font color="Green">E53</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:94_1_1_1_2_1"/>
<font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><a NAME="E2:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E9:94_1"><i><font color="Green">E52</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:94_1_1_1_2_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E131</font></i>: <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span>

;<br/><a NAME="E6:94_1_1_1_2_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="int_2.html#T4">INT_2:4</a></i>;<br/><a NAME="E7:94_1_1_1_2_1"/><b>then </b><i><font color="Green">E132</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E8:94_1_1_1_2_1"/><i><font color="Green">E134</font></i>: 
0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E11:94_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:94_1_1_1_2_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:94_1_1_1_2_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:94_1_1_1_2_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K3">|^</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K3">|^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="nat_3.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_3.html#T13" target="_self">Th13</a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
;<br/><a NAME="E16:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K3">|^</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E17:94_1_1_1_2_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:94_1_1_1_2_1_3"/>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:94_1_1_1_2_1_3"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="nat_3.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:94_1_1_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E19:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:94_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E21:94_1_1_1_2_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E22:94_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_3.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E23:94_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E24:94_1_1_1_2_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:94_1_1_1_2_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E28:94_1_1_1_2_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E29:94_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E30:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E31:94_1_1_1_2_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_2_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:94_1_1_1_2_1_4"/>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:94_1_1_1_2_1_4"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="nat_3.html#T16" target="_self">Th16</a>, <a class="ref" href="nat_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:94_1_1_1_2_1_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E33:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E35:94_1_1_1_2_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E36:94_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="nat_3.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E37:94_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E38:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E39:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/><a NAME="E40:94_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/><a NAME="E41:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E11:94_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E42:94_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E43:94_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E44:94_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:94_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K6" target="_self">max</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:94_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="xxreal_0.html#T1">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="E2:94"/>
 <a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">m</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="nat_3.html#K6" target="_self">max</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
