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<span class="kw">let </span><font color="Maroon" title="c2">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">l</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>   holds <br/>(  not 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l</font> or for <font color="Olive" title="b2">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/> for <font color="Olive" title="b3">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>   holds <br/>(  not  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">l</font> or  not  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">l</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83"/>
 ex <font color="Olive" title="b1">l</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>  st <br/>( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l</font> &amp;  ex <font color="Olive" title="b2">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b3">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>  st <br/>(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">l</font> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">l</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">l</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="E3:83"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">and </span><br/><a NAME="E4:83"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">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>  st <br/>(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">l</font> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">l</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">m</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c5">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="E6:83"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">and </span><br/><a NAME="E7:83"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E36">A5</font></span>: <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="rat_1.html#T37" target="_self" title="RAT_1:th.37">Th37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c4">m</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">k</font>

;<br/>
<a NAME="E9:83"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c5">k</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="#E7:83"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="rat_1.html#D3" target="_self" title="RAT_1:def.3">Def3</a></span>;<br/>
<a NAME="E10:83"/><span class="kw">then </span>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:83"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="rat_1.html#D3" target="_self" title="RAT_1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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