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<b>set </b><font color="Maroon">w</font> =  <a href="square_1.html#K7">sqrt</a> 2;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a>, <a class="ref" href="square_1.html#T84">SQUARE_1:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> 2
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2"/>
(  not <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> is <a href="rat_1.html#V1">irrational</a> or <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> is <a href="rat_1.html#V1">irrational</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1"><b>suppose </b></a><a NAME="E1:4_2_1"/><i><font color="Green">E41</font></i>: 
not <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> is <a href="rat_1.html#V1">irrational</a>
 ;<br/><div class="add"><b>take </b>
 <a href="square_1.html#K7">sqrt</a> 2
;<br/><b>take </b>
 <a href="square_1.html#K7">sqrt</a> 2
;<br/><b>thus </b><a NAME="E2:4_2_1"/>
(  <a href="square_1.html#K7">sqrt</a> 2 is <a href="rat_1.html#V1">irrational</a> &amp;  <a href="square_1.html#K7">sqrt</a> 2 is <a href="rat_1.html#V1">irrational</a> &amp; not <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> is <a href="rat_1.html#V1">irrational</a> )
 <b>by </b><i>, , <a class="ref" href="int_2.html#T44">INT_2:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_2"><b>suppose </b></a><a NAME="E1:4_2_2"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> is <a href="rat_1.html#V1">irrational</a>
 ;<br/><div class="add"><b>take </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span>
;<br/><b>take </b>
 <a href="square_1.html#K7">sqrt</a> 2
;<br/><b>thus </b><a NAME="E2:4_2_2"/>
( <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> is <a href="rat_1.html#V1">irrational</a> &amp;  <a href="square_1.html#K7">sqrt</a> 2 is <a href="rat_1.html#V1">irrational</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> is <a href="rat_1.html#V1">irrational</a> )
 <b>by </b><i>, , , <a class="ref" href="int_2.html#T44">INT_2:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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