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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:122"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:122"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font>, <font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:122"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:122"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E6:122"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<a NAME="E7:122"/>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/>
<a NAME="E8:122"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E9:122"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:122"/>
<font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<b>hence </b><a NAME="E11:122"/>
<font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="pepin.html#T34" target="_self">Th34</a>, <a class="ref" href="real_2.html#T144">REAL_2:144</a></i>;<br/>


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