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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <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> ;<br/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">nm</font> = <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">m</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E3:38"/>
( <font color="Maroon" title="c1">m</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">nm</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<a NAME="E4:38"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:38"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">nm</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T83" title="XREAL_1:th.83">XREAL_1:83</a></i>;<br/>
<b>hence </b><a NAME="E6:38"/>
<font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="catalan2.html#E1:38"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></i>;<br/>


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