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<span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">m1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">m1</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">m1</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">m1</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">m1</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_1"/>
( <font color="Maroon" title="c3">n</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>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_1_1"/>
<font color="Maroon" title="c3">n</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">m1</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">m1</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="lab"><font color="Green" title="E33">A3</font></span>: <font color="Maroon" title="c4">p</font> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></span>
;<br/><a NAME="E4:49_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:49_1_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E7:49_1_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="rat_1.html#T27" target="_self" title="RAT_1:th.27">Th60</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c7">y</font> =  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c6">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">y</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c2">m</font> = 
 <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#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">y</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c3">n</font> = 
 <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="c4">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">y</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><span class="kw">suppose </span></a><a NAME="E1:49_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">m1</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">m1</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:49_1_2"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E5:49_1_2"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="rat_1.html#T27" target="_self" title="RAT_1:th.27">Th60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">k</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">y</font> )</span><br/><span class="kw">thus </span><a NAME="E7:49_1_2"/>
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E5:49_1_2"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
