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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">take </span>
<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>
; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D4" target="_self" title="ARYTM_3:def.4">Def4</a></span>;<br/>
<a NAME="E3:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T7" target="_self" title="ARYTM_3:th.7">Th11</a></span>;<br/>
<a NAME="E4:18"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T12" target="_self" title="ARYTM_3:th.12">Th17</a></span>;<br/>
<a NAME="E5:18"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T7" target="_self" title="ARYTM_3:th.7">Th11</a></span>;<br/>
<a NAME="E6:18"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="ordinal3.html#T50" title="ORDINAL3:th.50">ORDINAL3:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="ordinal3.html#T33" title="ORDINAL3:th.33">ORDINAL3:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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