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<span class="kw">let </span><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#K5" title="ARYTM_3:func.5">RAT+</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <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>   st <font color="Olive" title="b1">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="Olive" title="b1">i</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  holds <br/> not <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> </span><br/><span class="kw">let </span><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#K5" title="ARYTM_3:func.5">RAT+</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="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#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  implies  not <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>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:93"/><span class="lab"><font color="Green" title="E75">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:93"/><span class="lab"><font color="Green" title="E76">A2</font></span>: 
<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#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">and </span><br/><a NAME="E3:93"/><span class="lab"><font color="Green" title="E77">A3</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">z</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:93"/><span class="lab"><font color="Green" title="E78">A4</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</a></span>;<br/>
<a NAME="E6:93"/>
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T64" target="_self" title="ARYTM_3:th.64">Th64</a></span>;<br/>
<a NAME="E7:93"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T37" target="_self" title="ARYTM_3:th.37">Th37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">z9</font> = <font color="Maroon" title="c5">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="txt" href="#E5:93"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="arytm_3.html#T78" target="_self" title="ARYTM_3:th.78">Th78</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E10:93"/><span class="lab"><font color="Green" title="E79">A5</font></span>: 
<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#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</a></span>;<br/>
<a NAME="E11:93"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T37" target="_self" title="ARYTM_3:th.37">Th37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">y9</font> = <font color="Maroon" title="c7">y</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="txt" href="#E10:93"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="ref" href="arytm_3.html#T78" target="_self" title="ARYTM_3:th.78">Th78</a></span>;<br/>
<a NAME="E13:93"/>
<font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:93"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="txt" href="#E5:93"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="arytm_3.html#T57" target="_self" title="ARYTM_3:th.57">Th57</a></span>;<br/>
<span class="kw">then </span>1 = 
<font color="Maroon" title="c7">y</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T69" target="_self" title="ARYTM_3:th.69">Th69</a></span>
<br/>.= 
<font color="Maroon" title="c8">y9</font> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <font color="Maroon" title="c6">z9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T64" target="_self" title="ARYTM_3:th.64">Th64</a></span>
;<br/>
<a NAME="E15:93"/><span class="kw">then </span>
<font color="Maroon" title="c8">y9</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></span>;<br/>
<a NAME="E16:93"/><span class="kw">then </span>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or <font color="Maroon" title="c7">y</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#T19" title="ORDINAL3:th.19">ORDINAL3:19</a></span>;<br/>
<a NAME="E17:93"/><span class="kw">then </span>
( <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#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:93"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:93"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E76">A2</font></span></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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