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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">c</font> implies <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font></span><br/>

<div><span class="lab"><font color="Green" title="E12">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font></span><br/><a NAME="E2:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">c</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="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></span>;<br/><a NAME="E3:10_1"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="int_2.html#T2" target="_self" title="INT_2:th.2">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:10_1"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="int_2.html#T1" target="_self" title="INT_2:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_2"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font></span><br/><a NAME="E2:10_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:10_2"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="int_2.html#E8"><span class="lab"><font color="Green" title="E8">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:10"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E12">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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