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<span class="kw">let </span><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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="c3">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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_1"/>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c5">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span><br/><div class="add"><a NAME="E2:56_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">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; <font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font> <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="arytm_3.html#D10" target="_self" title="ARYTM_3:def.10">Def10</a>, <a class="ref" href="ordinal2.html#T35" title="ORDINAL2:th.35">ORDINAL2:35</a></span>;<br/><span class="kw">hence </span><a NAME="E3:56_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D10" target="_self" title="ARYTM_3:def.10">Def10</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_1_2"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c5">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> 
 ; <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="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span><br/><div class="add"><a NAME="E2:56_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">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="#E1:56"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="ordinal3.html#T31" title="ORDINAL3:th.31">ORDINAL3:31</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A4</font></span>:  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> = 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T42" target="_self" title="ARYTM_3:th.42">Th48</a></span>
<br/>.= 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="arytm_3.html#T28" target="_self" title="ARYTM_3:th.28">Th33</a></span>
<br/>.= 
 <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_2"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="arytm_3.html#T42" target="_self" title="ARYTM_3:th.42">Th48</a></span>
;<br/> <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> = 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_2"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="arytm_3.html#T42" target="_self" title="ARYTM_3:th.42">Th48</a></span>
<br/>.= 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="arytm_3.html#T28" target="_self" title="ARYTM_3:th.28">Th33</a></span>
<br/>.= 
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_2"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="arytm_3.html#T42" target="_self" title="ARYTM_3:th.42">Th48</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_2"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="arytm_3.html#T39" target="_self" title="ARYTM_3:th.39">Th45</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T39" target="_self" title="ARYTM_3:th.39">Th45</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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