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<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> holds <br/><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )</span><br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:206_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_1"><span class="kw">suppose </span></a><a NAME="E1:206_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:206_1_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_2"><span class="kw">suppose </span></a><a NAME="E1:206_1_2"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )</span><br/><div class="add"><a NAME="E2:206_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" onmouseover="rs('newton/T4')" onmouseout="rh()">NEWTON:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:206_1_2"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T12" onmouseover="rs('int_2/T12')" onmouseout="rh()">INT_2:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:206_1_3"/><span class="lab"><font color="Green" title="E139">A1</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:206_1_3"/><span class="lab"><font color="Green" title="E140">A2</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:206_1_3"/><span class="lab"><font color="Green" title="E141">A3</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )</span><br/><div class="add"><span class="kw">assume </span><a NAME="E4:206_1_3"/><span class="lab"><font color="Green" title="E142">A4</font></span>: 
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:206_1_3"/>
<font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c8">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:206_1_3"><span class="lab"><font color="Green" title="E141">A3</font></span></a>, <a class="ref" href="newton02.html#T6" onmouseover="rs('newton02/T6')" onmouseout="rh()">NEWTON02:6</a></span>;<br/><a NAME="E6:206_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A5</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:206_1_3"><span class="lab"><font color="Green" title="E142">A4</font></span></a>, <a class="ref" href="nat_d.html#T4" onmouseover="rs('nat_d/T4')" onmouseout="rh()">NAT_D:4</a></span>;<br/><a NAME="E7:206_1_3"/><span class="lab"><font color="Green" title="E144">A6</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:206_1_3_1"/>
( ( <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> ) or (  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> ) or ( <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> ) or (  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:206_1_3_1_1"/>
( <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:206_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:206_1_3"><span class="lab"><font color="Green" title="E142">A4</font></span></a>, <a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:206_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T27" onmouseover="rs('int_2/T27')" onmouseout="rh()">INT_2:27</a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:206_1_3_1_2"/>
(  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:206_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:206_1_3"><span class="lab"><font color="Green" title="E144">A6</font></span></a>, <a class="ref" href="nat_3.html#T27" onmouseover="rs('nat_3/T27')" onmouseout="rh()">NAT_3:27</a></span>;<br/><a NAME="E3:206_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:206_1_3"><span class="lab"><font color="Green" title="E139">A1</font></span></a>, <a class="txt" href="#E2:206_1_3"><span class="lab"><font color="Green" title="E140">A2</font></span></a>, <a class="ref" href="nat_3.html#T28" onmouseover="rs('nat_3/T28')" onmouseout="rh()">NAT_3:28</a></span>;<br/><a NAME="E4:206_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:206_1_3"><span class="lab"><font color="Green" title="E139">A1</font></span></a>, <a class="txt" href="#E2:206_1_3"><span class="lab"><font color="Green" title="E140">A2</font></span></a>, <a class="txt" href="#E4:206_1_3"><span class="lab"><font color="Green" title="E142">A4</font></span></a>, <a class="ref" href="moebius2.html#T40" onmouseover="rs('moebius2/T40')" onmouseout="rh()">MOEBIUS2:40</a></span>;<br/><span class="kw">hence </span><a NAME="E5:206_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:206_1_3"><span class="lab"><font color="Green" title="E140">A2</font></span></a>, <a class="txt" href="#E4:206_1_3"><span class="lab"><font color="Green" title="E142">A4</font></span></a>, <a class="ref" href="moebius1.html#T9" onmouseover="rs('moebius1/T9')" onmouseout="rh()">MOEBIUS1:9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:206_1_3_1_3"/>
( <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:206_1_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:206_1_3"><span class="lab"><font color="Green" title="E144">A6</font></span></a>, <a class="ref" href="nat_3.html#T27" onmouseover="rs('nat_3/T27')" onmouseout="rh()">NAT_3:27</a></span>;<br/><a NAME="E3:206_1_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:206_1_3"><span class="lab"><font color="Green" title="E139">A1</font></span></a>, <a class="txt" href="#E2:206_1_3"><span class="lab"><font color="Green" title="E140">A2</font></span></a>, <a class="ref" href="nat_3.html#T28" onmouseover="rs('nat_3/T28')" onmouseout="rh()">NAT_3:28</a></span>;<br/><a NAME="E4:206_1_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:206_1_3"><span class="lab"><font color="Green" title="E139">A1</font></span></a>, <a class="txt" href="#E2:206_1_3"><span class="lab"><font color="Green" title="E140">A2</font></span></a>, <a class="txt" href="#E4:206_1_3"><span class="lab"><font color="Green" title="E142">A4</font></span></a>, <a class="ref" href="moebius2.html#T40" onmouseover="rs('moebius2/T40')" onmouseout="rh()">MOEBIUS2:40</a></span>;<br/><span class="kw">hence </span><a NAME="E5:206_1_3_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:206_1_3"><span class="lab"><font color="Green" title="E139">A1</font></span></a>, <a class="txt" href="#E4:206_1_3"><span class="lab"><font color="Green" title="E142">A4</font></span></a>, <a class="ref" href="moebius1.html#T9" onmouseover="rs('moebius1/T9')" onmouseout="rh()">MOEBIUS1:9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="206_1_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:206_1_3_1_4"/>
(  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp;  not <font color="Maroon" title="c8">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:206_1_3_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:206_1_3"><span class="lab"><font color="Green" title="E144">A6</font></span></a>, <a class="ref" href="nat_3.html#T27" onmouseover="rs('nat_3/T27')" onmouseout="rh()">NAT_3:27</a></span>;<br/><a NAME="E3:206_1_3_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:206_1_3"><span class="lab"><font color="Green" title="E139">A1</font></span></a>, <a class="txt" href="#E2:206_1_3"><span class="lab"><font color="Green" title="E140">A2</font></span></a>, <a class="ref" href="nat_3.html#T28" onmouseover="rs('nat_3/T28')" onmouseout="rh()">NAT_3:28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:206_1_3_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:206_1_3"><span class="lab"><font color="Green" title="E139">A1</font></span></a>, <a class="txt" href="#E2:206_1_3"><span class="lab"><font color="Green" title="E140">A2</font></span></a>, <a class="txt" href="#E6:206_1_3"><span class="lab"><font color="Green" title="E143">A5</font></span></a>, <a class="txt" href="#E7:206_1_3"><span class="lab"><font color="Green" title="E144">A6</font></span></a>, <a class="ref" href="nat_3.html#T27" onmouseover="rs('nat_3/T27')" onmouseout="rh()">NAT_3:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
