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<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/><b>let </b><font color="Maroon" title="c4">i</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#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>



<b>assume </b><a NAME="E1:97"/><i><font color="Green" title="E80">A1</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 ;<br/>

<a NAME="E2:97"/><i><font color="Green" title="E81">A2</font></i>: 
<font color="Maroon" title="c4">i</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="ref" href="arytm_3.html#T37" target="_self" title="ARYTM_3:th.37">Th37</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:97"/><i><font color="Green" title="E82">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:97"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E6:97"/><i><font color="Green" title="E83">A4</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:97"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</a></i>;<br/>
<a NAME="E7:97"/>
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> 1
 
<b>by </b><i><a class="ref" href="arytm_3.html#T64" target="_self" title="ARYTM_3:th.64">Th64</a></i>;<br/>
<a NAME="E8:97"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <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="ref" href="arytm_3.html#T37" target="_self" title="ARYTM_3:th.37">Th37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c6">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="arytm_3.html#E1:97"><i><font color="Green" title="E80">A1</font></i></a>, <a class="txt" href="arytm_3.html#E2:97"><i><font color="Green" title="E81">A2</font></i></a>, <a class="txt" href="arytm_3.html#E4:97"><i><font color="Green" title="E82">A3</font></i></a>, <a class="txt" href="arytm_3.html#E6:97"><i><font color="Green" title="E83">A4</font></i></a>, <a class="ref" href="arytm_3.html#T78" target="_self" title="ARYTM_3:th.78">Th78</a></i>;<br/>
<a NAME="E10:97"/>
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c6">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E4:97"><i><font color="Green" title="E82">A3</font></i></a>, <a class="txt" href="arytm_3.html#E6:97"><i><font color="Green" title="E83">A4</font></i></a>, <a class="ref" href="arytm_3.html#T57" target="_self" title="ARYTM_3:th.57">Th57</a></i>;<br/>
<b>then </b>1 = 
<font color="Maroon" title="c5">y</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T69" target="_self" title="ARYTM_3:th.69">Th69</a></i>
<br/>.= 
<font color="Maroon" title="c7">y'</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> <font color="Maroon" title="c8">z'</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T64" target="_self" title="ARYTM_3:th.64">Th64</a></i>
;<br/>
<a NAME="E12:97"/><b>then </b>
( <font color="Maroon" title="c7">y'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 1 &amp; <font color="Maroon" title="c8">z'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 1 )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></i>;<br/>
<a NAME="E13:97"/><b>then </b>
( ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T19" title="ORDINAL3:th.19">ORDINAL3:19</a></i>;<br/>
<a NAME="E14:97"/><b>then </b>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E4:97"><i><font color="Green" title="E82">A3</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></i>;<br/>
<b>hence </b><a NAME="E15:97"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:97"><i><font color="Green" title="E80">A1</font></i></a></i>;<br/>


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