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

<b>let </b><font color="Maroon" title="c2">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> ;<br/>

<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">b</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>  ;<br/>
<b>let </b><font color="Maroon" title="c7">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<div><i><font color="Green" title="E41">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1"/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E2:54_1"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>;<br/><a NAME="E3:54_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><a NAME="E4:54_1"/><b>then </b><i><font color="Green" title="E43">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:54_1"/><i><font color="Green" title="E44">A5</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></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">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="int_3.html#E2:54_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></i>;<br/><a NAME="E7:54_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</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> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 ;<br/><b>hence </b><a NAME="E8:54_1"/>
( <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</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> <font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E4:54_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="int_3.html#E6:54_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_2"/><i><font color="Green" title="E42">A6</font></i>: 
( <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</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> <font color="Maroon" title="c2">n</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c8">c</font> being    <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> such that </b><br/><a NAME="E3:54_2"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 ;<br/><b>consider </b><font color="Maroon" title="c9">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:54_2"/><i><font color="Green" title="E44">A8</font></i>: 
( ( <font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> ) or ( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 <b>by </b><i><a class="txt" href="int_3.html#E3:54_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c10">q</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:54_2_1_1"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">q</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:54_2"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E3:54_2_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_2_1_1_1_1_1"/>
( <font color="Maroon" title="c9">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c9">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:54_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">r</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:54_2_1_1_1_1_1_1"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c9">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E4:54_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="int_3.html#E5:54_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="int_3.html#E2:54_2_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="int_3.html#E3:54_2_1_1_1_1_1_1"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/><a NAME="E5:54_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">q</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c9">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:54_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:54_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><a NAME="E2:54_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_2_1_1_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/>

<a NAME="E2:54_2_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1_1_1_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_2_1_1_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:54_2_1_1_1_1_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E3:54_2_1_1_1_1_1_1_1_1_2_1_1"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">r</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E4:54_2_1_1_1_1_1_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_3.html#E1:54_2_1_1_1_1_1_1_1_1_2_1"><i><font color="Green" title="E49">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_2_1_1_1_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_3.html#E3:54_2_1_1_1_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="int_3.html#E1:54_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E48">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:54_2_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/><a NAME="E4:54_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">q</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:54_2_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E6:54_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="int_3.html#E5:54_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="int_3.html#E5:54_2_1_1_1_1_1_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="int_3.html#E4:54_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:54_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:54_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><a NAME="E2:54_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">t</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E3:54_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">t</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/><a NAME="E4:54_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E5:54_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:54_2"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="int_3.html#E5:54_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="int_3.html#E3:54_2_1_1_1_1_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_2_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:54_2_1_1"/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="int_3.html#E3:54_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="int_3.html#E5:54_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:54_2_1"/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:54_2"/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:54"/>
( ( <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</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> <font color="Maroon" title="c2">n</font> ) iff <span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="int_3.html#E3:54"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>


</div>
