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<div class="add">

<b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">p</font> iff (  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> ) )</span><br/>



<div><i><font color="Green" title="E41">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1"/><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> )</span><br/><b>set </b><font color="Maroon" title="c4">r</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">p</font>;<br/><a NAME="E2:66_1"/><i><font color="Green" title="E43">A3</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:66_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><b>set </b><font color="Maroon" title="c5">s</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">q</font>;<br/><a NAME="E3:66_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:66_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T189" title="XCMPLX_1:th.189">XCMPLX_1:189</a></i>;<br/><a NAME="E4:66_1"/><b>then </b>
(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rat_1.html#E2:66_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="rat_1.html#T88" target="_self" title="RAT_1:th.88">Th88</a></i>;<br/><a NAME="E5:66_1"/><b>then </b>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rat_1.html#T87" target="_self" title="RAT_1:th.87">Th87</a></i>;<br/><a NAME="E6:66_1"/><b>then </b>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rat_1.html#T87" target="_self" title="RAT_1:th.87">Th87</a></i>;<br/><b>hence </b><a NAME="E7:66_1"/>
(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2"/><i><font color="Green" title="E42">A4</font></i>: 
(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">p</font> )</span><br/><a NAME="E2:66_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="rat_1.html#T27" target="_self" title="RAT_1:th.27">Th27</a></i>;<br/><a NAME="E3:66_2"/><b>then </b>
 <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><b>hence </b><a NAME="E4:66_2"/>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rat_1.html#T76" target="_self" title="RAT_1:th.76">Th76</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">p</font></span><br/><b>thus </b><font color="Maroon" title="c3">q</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E1:66_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="rat_1.html#T37" target="_self" title="RAT_1:th.37">Th37</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T189" title="XCMPLX_1:th.189">XCMPLX_1:189</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span>

<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="rat_1.html#T37" target="_self" title="RAT_1:th.37">Th37</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
( <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">p</font> iff (  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="rat_1.html#E1:66"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
