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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c3">n2</font>, <font color="Maroon" title="c4">m2</font> be    <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> ;<br/>



<b>thus </b><a NAME="E1:13"/>
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> implies ( <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="int_1.html#T75" title="INT_1:th.75">INT_1:75</a>, <a class="ref" href="nat_d.html#D1" title="NAT_D:def.1">NAT_D:def 1</a></i>;<br/>

<b>thus </b><a NAME="E2:13"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font> )
 <b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>

<b>thus </b><a NAME="E3:13"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> implies ( ( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> ) &amp; ( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> )
 ;<br/>

<b>thus </b><a NAME="E2:13_1"/>
( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font>
 ;<br/>

<a NAME="E2:13_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/>
<a NAME="E3:13_1_1"/><b>then </b>
<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E4:13_1_1"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">n2</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_1_1"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">m</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T192" title="XCMPLX_1:th.192">XCMPLX_1:192</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">m2</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:13_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="prgcor_1.html#E4:13_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="jordan1d.html#T6" title="JORDAN1D:th.6">JORDAN1D:6</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:13_1"/>
( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_2"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n2</font>
 ;<br/>

<a NAME="E2:13_1_2"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/>
<a NAME="E3:13_1_2"/><b>then </b>
<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E4:13_1_2"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
not <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_1_2"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<i><font color="Green" title="E16">A8</font></i>: <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">m</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T192" title="XCMPLX_1:th.192">XCMPLX_1:192</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">m2</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
;<br/>
<a NAME="E6:13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:13_1_2"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="prgcor_1.html#E4:13_1_2"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="jordan1d.html#T5" title="JORDAN1D:th.5">JORDAN1D:5</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="prgcor_1.html#E5:13_1_2"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:13"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> implies ( ( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> ) &amp; ( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 ;<br/>

<b>thus </b><a NAME="E2:13_2"/>
( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2_1"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font>
 ;<br/>

<a NAME="E2:13_2_1"/>
<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E3:13_2_1"/><b>then </b><i><font color="Green" title="E14">A11</font></i>: 
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">n2</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2_1"><i><font color="Green" title="E13">A10</font></i></a></i>;<br/>
<font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2"><i><font color="Green" title="E12">A9</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="prgcor_1.html#E3:13_2_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="jordan1d.html#T6" title="JORDAN1D:th.6">JORDAN1D:6</a></i>
;<br/>
<b>hence </b><a NAME="E5:13_2_1"/>
<font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:13_2"/>
( <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n2</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2_2"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n2</font>
 ;<br/>

<a NAME="E2:13_2_2"/>
<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E3:13_2_2"/><b>then </b>
not <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2_2"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/>
<a NAME="E4:13_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="jordan1d.html#T5" title="JORDAN1D:th.5">JORDAN1D:5</a></i>;<br/>
<b>hence </b><a NAME="E5:13_2_2"/>
<font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_2"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:13"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3"/><i><font color="Green" title="E12">A13</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> )
 ;<br/>

<b>thus </b><font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">m</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T192" title="XCMPLX_1:th.192">XCMPLX_1:192</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:13_3"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
;<br/>


</div><b>end;</b></div>


</div>
