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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><br/>



<a NAME="E1:51"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</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">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T1" title="ABSVALUE:th.1">ABSVALUE:1</a></span>;<br/>
<a NAME="E2:51"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</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">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T1" title="ABSVALUE:th.1">ABSVALUE:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E32">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">m</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"> (  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <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">m</font> &amp;  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</font> implies <font color="Maroon" title="c1">a</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:51_1"/>
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <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">m</font> &amp;  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</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_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</font></span><br/><a NAME="E2:51_1"/><span class="kw">then </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">m</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="int_2.html#T14" target="_self" title="INT_2:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</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>
<a NAME="E4:51"/>
<font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</a></span>;<br/>
<a NAME="E5:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="int_2.html#T14" target="_self" title="INT_2:th.14">Th14</a></span>;<br/>
<a NAME="E6:51"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</a></span>;<br/>
<a NAME="E7:51"/><span class="kw">then </span>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="int_2.html#T14" target="_self" title="INT_2:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:51"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</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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