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<b>let </b><font color="Maroon" title="c1">c</font>, <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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">c</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  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )</span><br/>





<a NAME="E1:33"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#D5" target="_self" title="ARYTM_3:def.5">Def5</a></i>;<br/>
<a NAME="E2:33"/><i><font color="Green" title="E34">A2</font></i>: 
( <font color="Maroon" title="c2">a</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 <font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</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>  )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T23" target="_self" title="ARYTM_3:th.23">Th23</a></i>;<br/>
<b>assume </b><a NAME="E3:33"/>
<font color="Maroon" title="c1">c</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">  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c2">a</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">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</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>  )
 
<b>by </b><i><a class="txt" href="#E2:33"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green" title="E36">A4</font></i>: 
(  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a>, <a class="ref" href="ordinal3.html#T83" title="ORDINAL3:th.83">ORDINAL3:83</a></i>;<br/>
<b>thus </b> <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T22" target="_self" title="ARYTM_3:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">c</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></i>
<br/>.= 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="#E4:33"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E5:33"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="ordinal3.html#T81" title="ORDINAL3:th.81">ORDINAL3:81</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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