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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</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">i</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c1">i</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">j</font> )</span><br/>

<b>assume </b><a NAME="E1:15"/>
<font color="Maroon" title="c1">i</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">j</font></span><br/>

<b>then consider </b><font color="Maroon" title="c3">C</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:15"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T30" title="ORDINAL3:th.30">ORDINAL3:30</a></i>;<br/>
<a NAME="E4:15"/>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="ordinal3.html#T87" title="ORDINAL3:th.87">ORDINAL3:87</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">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="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <b>by </b><i><a class="txt" href="numbers.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E6:15"/>
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="arytm_3.html#T64" title="ARYTM_3:th.64">ARYTM_3:64</a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon" title="c1">i</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</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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