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





<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:38"/>
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T23" target="_self" title="ARYTM_3:th.23">Th23</a></span>;<br/>
<a NAME="E3:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A2</font></span>: 
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></span>;<br/>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
(  <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>  )
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E5:38"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D5" target="_self" title="ARYTM_3:def.5">Def5</a></span>;<br/>
<span class="kw">thus </span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T22" target="_self" title="ARYTM_3:th.22">Th22</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="ordinal3.html#T81" title="ORDINAL3:th.81">ORDINAL3:81</a>, <a class="ref" href="ordinal3.html#T83" title="ORDINAL3:th.83">ORDINAL3:83</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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