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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:81"/>
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span></span><br/>

<a NAME="E2:81"/><b>then </b><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font>;<br/>
<b>set </b><font color="Maroon" title="c4">g</font> =  <a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font>;<br/>
<b>assume </b><a NAME="E3:81"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:81"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>and </b><br/><a NAME="E6:81"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> <b>by </b><i><a class="txt" href="#E5:81"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="nat_3.html#T34" target="_self" title="NAT_3:th.34">Th34</a></i>;<br/>
<a NAME="E8:81"/><i><font color="Green" title="E48">A5</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E5:81"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E9:81"/><i><font color="Green" title="E49">A6</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></i>;<br/>
<a NAME="E10:81"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E11:81"/><b>then </b><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/>
<a NAME="E12:81"/>
<font color="Maroon" title="c6">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">x</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E8:81"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E9:81"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="nat_3.html#T3" target="_self" title="NAT_3:th.3">Th3</a></i>;<br/>
<a NAME="E13:81"/><b>then </b><i><font color="Green" title="E52">A9</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="#E11:81"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/>
<a NAME="E14:81"/><i><font color="Green" title="E53">A10</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E6:81"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E15:81"/><i><font color="Green" title="E54">A11</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></i>;<br/>
<a NAME="E16:81"/><i><font color="Green" title="E55">A12</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="#E10:81"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/>
<a NAME="E17:81"/>
<font color="Maroon" title="c6">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">x</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:81"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="#E15:81"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="nat_3.html#T3" target="_self" title="NAT_3:th.3">Th3</a></i>;<br/>
<a NAME="E18:81"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="#E16:81"><i><font color="Green" title="E55">A12</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/>
<a NAME="E19:81"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 
<b>by </b><i><a class="txt" href="#E2:81"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="#E13:81"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<b>hence </b><a NAME="E20:81"/>
contradiction
 <b>by </b><i><a class="txt" href="#E10:81"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="wsierp_1.html#T20" title="WSIERP_1:th.20">WSIERP_1:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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