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<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">a</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>set </b><font color="Maroon">x</font> = <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E1:61"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E2:61"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<a NAME="E3:61"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:61"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:61"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</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">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:61_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E3:61_1"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:61_1"/><i><font color="Green">E69</font></i>: 
not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, </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="E6:61_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma33</font></i></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="E8:61_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span> <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">v</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:61_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E105</font></i>: <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
<b>by </b><i>, <a class="txt" href="nat_3.html#E1:61"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></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>

;<br/><a NAME="E12:61_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:61_1"/><b>then </b>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">v</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="txt" href="nat_3.html#E2:61"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E14:61_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">p</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="hsNdiv(this)" href="javascript:()" title="61_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_2"/>
( <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font> or <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u</font> )
 </span><b>by </b><i><a class="ref" href="nat_3.html#T2" target="_self">Th2</a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1"><b>suppose </b></a><a NAME="E1:61_1_2_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:61_1_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t1</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</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">t1</font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:61_1_2_1"/>
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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2"><b>suppose </b></a><a NAME="E1:61_1_2_2"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:61_1_2_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t1</font>
<b>by </b><i><a class="txt" href="nat_3.html#E1:61"><i><font color="Green">E41</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</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">t1</font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:61_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_3.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:61"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
