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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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) iff <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<a NAME="E1:14"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1"/>
<font color="Maroon" title="c2">b</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_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:14_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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/>
<span class="kw">hence </span><a NAME="E3:14_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="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_2.html#T3" target="_self" title="INT_2:th.3">Th3</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="E2:14"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
(  not <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:14_2"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
( <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T2" target="_self" title="INT_2:th.2">Th2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:14_2"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<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>  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E3:14_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="c1">a</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="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T2" target="_self" title="INT_2:th.2">Th2</a></span>;<br/>
<a NAME="E4:14_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_2"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E1:14_2"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</a></span>;<br/>
<a NAME="E5:14_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</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_2.html#T3" target="_self" title="INT_2:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:14_2"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</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="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</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="E3:14"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_3"/>
<font color="Maroon" title="c1">a</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_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:14_3"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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/>
<span class="kw">hence </span><a NAME="E3:14_3"/>
<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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T3" target="_self" title="INT_2:th.3">Th3</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="E4:14"/>
( ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) iff <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</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>
