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

<b>let </b><font color="Maroon" title="c1">i2</font>, <font color="Maroon" title="c2">i3</font>, <font color="Maroon" title="c3">i1</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c1">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">i3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( <font color="Maroon" title="c2">i3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c2">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="radix_1.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c2">i3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:7_1_1"/><b>then </b>
<font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="radix_1.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font>
<b>by </b><i><a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font>

;<br/><b>hence </b><a NAME="E4:7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font>
 <b>by </b><i><a class="ref" href="euler_1.html#T13" title="EULER_1:th.13">EULER_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c2">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i3</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">i3</font>
 <b>by </b><i><a class="txt" href="radix_1.html#E1:7_1_2"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
