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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<a NAME="E2:3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</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/>
<b>assume </b><a NAME="E3:3"/>
not  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">p</font> is <a href="rat_1.html#V1">irrational</a>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a><br/> for <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">i1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Olive">n1</font> holds <br/><font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n1</font> ) )
 <b>by </b><i><a class="ref" href="rat_1.html#T25">RAT_1:25</a></i>;<br/>
<a NAME="E6:3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E7:3"/>
(  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E8:3"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<i><font color="Green">E46</font></i>: <font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">p</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
;<br/>
<a NAME="E11:3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E12:3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/>
<b>then consider </b><font color="Maroon">m1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</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> <a href="pepin.html#K2">^2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p</font>
<b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="pepin.html#K2">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
;<br/>
<a NAME="E16:3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E17:3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E20:3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">m1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , , , <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<a NAME="E23:3"/>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<b>hence </b><a NAME="E24:3"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/>


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