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<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K13">{}</a> ]
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a>  holds <br/><font color="Olive">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:68"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">a</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<a NAME="E1:68_1"/>
 <a href="arytm_3.html#K14">one</a>  <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">rone</font> =  <a href="arytm_3.html#K14">one</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  <b>by </b><i/>;<br/>
<a NAME="E3:68_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:68_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:68_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E8:68_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E9:68_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">rone</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E10:68_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">rone</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<font color="Maroon">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y'</font> = 
<font color="Maroon">i</font> <a href="arytm_3.html#K2">+^</a> 1
<b>by </b><i><a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma66</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>
;<br/>
<b>hence </b><a NAME="E13:68_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, , <a class="txt" href="arytm_2.html#E28"><i><font color="Green">Lemma68</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:68"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>]
 
<b>from </b><i><a class="ref" href="arytm_3.html#S1">ARYTM_3:sch 1</a>(<a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:68"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<a NAME="E6:68"/><b>then </b>
<font color="Maroon">e</font> is  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:68"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


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