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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">m</font> )</span><br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">m</font></span><br/>

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">ARYTM_3:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/>

<a NAME="E2:18"/>
( <font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#D4" target="_self" title="ARYTM_3:def.4">Def4</a>, <a class="ref" href="arytm_3.html#T17" target="_self" title="ARYTM_3:th.17">Th17</a></i>;<br/>
<a NAME="E3:18"/><b>then </b>
( <font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</a></i>;<br/>
<a NAME="E4:18"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></i>;<br/>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="ordinal3.html#T36" title="ORDINAL3:th.36">ORDINAL3:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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