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<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>

<b>assume </b><a NAME="E1:50"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>(  <a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:50"/><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>(  <a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:50"/><i><font color="Green">E33</font></i>: 
(  <a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rat_1.html#E3:50"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green">E34</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rat_1.html#E3:50"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="rat_1.html#E5:50"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rat_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E8:50"/><b>then </b><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><a NAME="E1:50_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E3:50"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rat_1.html#E8:50"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E2:50_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E5:50"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E5:50"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rat_1.html#T37" target="_self">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1
<b>by </b><i><a class="txt" href="rat_1.html#E6:50"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font>

;<br/>
<b>hence </b><a NAME="E11:50"/>
contradiction
 <b>by </b><i><a class="txt" href="rat_1.html#E7:50"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rat_1.html#E9:50"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rat_1.html#D3" target="_self">Def3</a></i>;<br/>


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