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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>,  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="ordinal1.html#NM2">Ordinal</a> = <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>,  <a href="ordinal2.html#NM1">Ordinal-Sequence</a>) <b>-&gt; </b>  <a href="ordinal1.html#NM2">Ordinal</a> =  <a href="ordinal2.html#K8" target="_self">lim</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:61_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font>) ) &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>3</sub></font>) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st ex <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font>) ) &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>4</sub></font>) ) ) &amp; ex <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font>) ) &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>4</sub></font>) ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>from </b><i><a class="ref" href="ordinal2.html#S13" target="_self">ORDINAL2:sch 13</a>();<br/></i>
<b>hence </b><a NAME="E2:61_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st ex <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) ) &amp; ex <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>


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