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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="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">a</font>,<font color="Maroon" title="c2">b</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  iff <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>



<a NAME="E1:26"/>
<font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">b</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/>
<a NAME="E2:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</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="E3:26"/>
<font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">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/>
<a NAME="E4:26"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</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/>
<span class="kw">hence </span><a NAME="E5:26"/>
( <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  implies <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="arytm_3.html#D2" target="_self" title="ARYTM_3:def.2">Def2</a></span>; <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">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E6:26"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">a</font>,<font color="Maroon" title="c2">b</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="arytm_3.html#D2" target="_self" title="ARYTM_3:def.2">ARYTM_3:def 2</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c4">d1</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c5">d2</font> implies <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E7:26"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c4">d1</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c5">d2</font> )
 ; <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="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E8:26"/>
( <font color="Maroon" title="c1">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>  or <font color="Maroon" title="c2">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="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="arytm_3.html#T14" target="_self" title="ARYTM_3:th.14">Th19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c3">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="ordinal3.html#T75" title="ORDINAL3:th.75">ORDINAL3:75</a></span>;<br/>
<a NAME="E10:26"/>
( <font color="Maroon" title="c6">c</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c6">c</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a></span>;<br/>
<a NAME="E11:26"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="arytm_3.html#D5" target="_self" title="ARYTM_3:def.5">Def5</a></span>;<br/>
<a NAME="E12:26"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">d</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b1">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T5" target="_self" title="ARYTM_3:th.5">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:26"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T37" title="ORDINAL3:th.37">ORDINAL3:37</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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