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<b>let </b><font color="Maroon">i2</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">i1</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:82"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
( <font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:82_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i2</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E3:82_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="int_1.html#K5">div</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>;<br/><a NAME="E4:82_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="int_1.html#K5">div</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:82_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="int_1.html#K5">div</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E6:82_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="int_1.html#K5">div</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>hence </b><a NAME="E7:82_1_1"/>
<font color="Maroon">i1</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_2"/>
<font color="Maroon">i1</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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