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<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ;<br/>

<a NAME="E2:3"/><b>then </b><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>assume </b><a NAME="E3:3"/>
not  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">p</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:3"/><i><font color="Green" title="E4">A3</font></i>: 
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font> &amp; ( for <font color="Olive" title="b1">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/> for <font color="Olive" title="b2">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">n1</font> holds <br/><font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n1</font> ) )
 <b>by </b><i><a class="ref" href="rat_1.html#T25" title="RAT_1:th.25">RAT_1:25</a></i>;<br/>
<a NAME="E6:3"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="irrat_1.html#E5:3"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E7:3"/>
(  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E8:3"/><b>then </b>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="irrat_1.html#E6:3"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c2">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<i><font color="Green" title="E6">A5</font></i>: <font color="Maroon" title="c4">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E6:3"><i><font color="Green" title="E5">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>
;<br/>
<a NAME="E11:3"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E12:3"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="txt" href="irrat_1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:3"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">m1</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<font color="Maroon" title="c3">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="irrat_1.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="irrat_1.html#E10:3"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="irrat_1.html#E14:3"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
;<br/>
<a NAME="E16:3"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E17:3"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="irrat_1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:3"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E20:3"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:3"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c6">n1</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="irrat_1.html#E5:3"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="irrat_1.html#E19:3"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>
<a NAME="E22:3"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c5">m1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="irrat_1.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="irrat_1.html#E5:3"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="irrat_1.html#E14:3"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="irrat_1.html#E19:3"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/>
<a NAME="E23:3"/>
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n1</font>
 
<b>by </b><i><a class="txt" href="irrat_1.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="irrat_1.html#E21:3"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="real_2.html#T199" title="REAL_2:th.199">REAL_2:199</a></i>;<br/>
<b>hence </b><a NAME="E24:3"/>
contradiction
 <b>by </b><i><a class="txt" href="irrat_1.html#E5:3"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="irrat_1.html#E19:3"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="irrat_1.html#E20:3"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="irrat_1.html#E21:3"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="irrat_1.html#E22:3"><i><font color="Green" title="E11">A10</font></i></a></i>;<br/>


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