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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">nm</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/>
<a NAME="E3:38"/>
( <font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">nm</font> <a href="newton.html#K6">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
<a NAME="E4:38"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K6">!</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> 0 &amp; <font color="Maroon">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K6">!</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a>, <a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
<a NAME="E5:38"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">nm</font> <a href="newton.html#K6">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T83">XREAL_1:83</a></i>;<br/>
<b>hence </b><a NAME="E6:38"/>
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/>


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