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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">m</font>, <font color="Olive" title="b2">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> implies <font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:121"/><span class="lab"><font color="Green" title="E92">A1</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:121"/><span class="lab"><font color="Green" title="E93">A2</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c3">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E4:121"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E5:121"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A3</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/>
<a NAME="E6:121"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/>
<a NAME="E7:121"/><span class="kw">then </span>
<font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="ref" href="pepin.html#T70" target="_self" title="PEPIN:th.70">Th70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:121"/>
<font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="txt" href="#E5:121"><span class="lab"><font color="Green" title="E94">A3</font></span></a>, <a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a>, <a class="ref" href="xreal_1.html#T157" title="XREAL_1:th.157">XREAL_1:157</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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