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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">T</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/><b>let </b><font color="Maroon">O</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice5.html#K6" target="_self">new_set</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="ordinal1.html#NM3">T-Sequence</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">O</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">O</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">O1</font>)
 ;<br/>

<a NAME="E4:48"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">O</font>) iff  ex <font color="Olive">L0</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1">last</a> <font color="Olive">L0</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">L0</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">O</font> &amp; <font color="Olive">L0</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">O</font> holds <br/><font color="Olive">L0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">C</font>,<font color="Olive">L0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font>) ) &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">O</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">L0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">C</font>,<font color="Olive">L0</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font>) ) ) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:48"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">O</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">O</font>,<font color="Maroon">T</font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S10">ORDINAL2:sch 10</a>(, , , <a class="txt" href="lattice5.html#E2:48"><i><font color="Green">E37</font></i></a>);</i><br/>


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