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<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> ]
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E63">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<a NAME="E3:68"/><i><font color="Green" title="E64">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">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" title="c5">a</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<a NAME="E1:68_1"/>
 <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="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">rone</font> =  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></i>;<br/>
<a NAME="E3:68_1"/><i><font color="Green" title="E65">A4</font></i>: 
<font color="Maroon" title="c5">a</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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:68_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E11"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></i>;<br/>
<b>assume </b><a NAME="E6:68_1"/><i><font color="Green" title="E66">A5</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">x'</font>, <font color="Maroon" title="c9">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="E8:68_1"/><i><font color="Green" title="E67">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font>
 <b>and </b><br/><a NAME="E9:68_1"/><i><font color="Green" title="E68">A7</font></i>: 
<font color="Maroon" title="c6">rone</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y'</font>
 <b>and </b><br/><a NAME="E10:68_1"/><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c6">rone</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c9">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:68_1"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="arytm_2.html#E77"><i><font color="Green" title="E61">Lm48</font></i></a>, <a class="txt" href="arytm_2.html#E11"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c5">a</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<font color="Maroon" title="c8">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c9">y'</font> = 
<font color="Maroon" title="c10">i</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> 1
<b>by </b><i><a class="txt" href="arytm_2.html#E8:68_1"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="arytm_2.html#E9:68_1"><i><font color="Green" title="E68">A7</font></i></a>, <a class="txt" href="arytm_2.html#E75"><i><font color="Green" title="E59">Lm46</font></i></a></i>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>
;<br/>
<b>hence </b><a NAME="E13:68_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:68"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="arytm_2.html#E6:68_1"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="arytm_2.html#E10:68_1"><i><font color="Green" title="E69">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:68"/><i><font color="Green" title="E65">A9</font></i>: 
for <font color="Olive" title="b1">a</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 
<b>from </b><i><a class="ref" href="arytm_3.html#S1" title="ARYTM_3:sch.1">ARYTM_3:sch 1</a>(<a class="txt" href="arytm_2.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="arytm_2.html#E3:68"><i><font color="Green" title="E64">A3</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:68"/>
<font color="Maroon" title="c5">e</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="E6:68"/><b>then </b>
<font color="Maroon" title="c5">e</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:68"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:68"><i><font color="Green" title="E65">A9</font></i></a></i>;<br/>


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