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<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Maroon">c<sub>3</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> or  not <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>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:51"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="rat_1.html#E1:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rat_1.html#E3:51"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:51"/>
<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#E1:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rat_1.html#E3:51"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E6:51"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E7:51"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:51"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="rat_1.html#E4:51"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E9:51"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E3:51"><i><font color="Green">E34</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:51"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E3:51"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rat_1.html#E8:51"><i><font color="Green">E36</font></i></a></i>;<br/>


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