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<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:23"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="card_4.html#T29" target="_self">Th29</a></i>;<br/>
<b>assume </b><a NAME="E3:23"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/>

<a NAME="E4:23"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E2:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal3.html#T48">ORDINAL3:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E4:23_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E3:23_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="card_1.html#T52">CARD_1:52</a></i>;<br/><a NAME="E5:23_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>;<br/><b>hence </b><a NAME="E6:23_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_4.html#E4:23"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:23"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E2:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/>


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