<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">t</font>, <font color="Maroon" title="c2">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">t</font> or <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">t</font> or <font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">t</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">t</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:35_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">s</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c3">m1</font> = <font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">s</font> as    <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>  <b>by </b><i><a class="txt" href="newton.html#E1:35_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E2:35_1_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">t</font>
 
<b>by </b><i><a class="txt" href="newton.html#E1:35_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">m2</font> = <font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E4:35_1_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E2:35_1_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a></i>;<br/>
<a NAME="E5:35_1_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="newton.html#T25" target="_self" title="NEWTON:th.25">Th25</a></i>;<br/>
<a NAME="E6:35_1_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23" target="_self" title="NEWTON:th.23">Th23</a></i>;<br/>
<a NAME="E7:35_1_1_1"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23" target="_self" title="NEWTON:th.23">Th23</a></i>;<br/>
<i><font color="Green" title="E28">A7</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="newton.html#E1:35_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="newton.html#E4:35_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="newton.html#E5:35_1_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T117" title="XCMPLX_1:th.117">XCMPLX_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21" target="_self" title="NEWTON:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="newton.html#E6:35_1_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21" target="_self" title="NEWTON:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="newton.html#E7:35_1_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21" target="_self" title="NEWTON:th.21">Th21</a></i>
;<br/>
<a NAME="E9:35_1_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E1:35_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E11:35_1_1_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m1</font>
 
;<br/>
<b>hence </b><a NAME="E12:35_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="newton.html#E8:35_1_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="newton.html#E9:35_1_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">t</font>
 ;<br/><div class="add"><a NAME="E2:35_1_2"/><b>then </b>
<font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:35_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="newton.html#E1:35_1_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="newton.html#T31" target="_self" title="NEWTON:th.31">Th31</a>, <a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:35_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">s</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_3"><b>suppose </b></a><a NAME="E1:35_1_3"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">t</font>
 ;<br/><div class="add"><a NAME="E2:35_1_3"/><b>then </b>
( <font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 &amp; <font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E3:35_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="newton.html#E1:35_1_3"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:35_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">s</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
