<?xml version="1.0"?>
<div class="add">

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





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>;<br/>
<a NAME="E1:48"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> = 
<font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:48"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> = 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:48"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">fi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E3:48_3"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font>;<br/>
<b>consider </b><font color="Maroon">psi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E5:48_3"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E6:48_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font> &amp; <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:48_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal3.html#E3:48_3"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="ordinal2.html#T46">ORDINAL2:46</a></i>;<br/>
<a NAME="E7:48_3"/><i><font color="Green">E40</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal3.html#K1" target="_self">+^</a> <font color="Maroon">psi</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal3.html#T10" target="_self">Th10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_3_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">n</font> = <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">m</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E3:48_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal3.html#E1:48_3_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal3.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">k</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E3:48_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal3.html#E1:48_3_1"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:48_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal3.html#E1:48_3_1"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E5:48_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ordinal3.html#E1:48_3_1"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal3.html#K1" target="_self">+^</a> <font color="Maroon">psi</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E5:48_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ordinal3.html#E1:48_3_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal3.html#T9" target="_self">Th9</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E9:48_3"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal3.html#K1" target="_self">+^</a> <font color="Maroon">psi</font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E3:48_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal3.html#E5:48_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ordinal3.html#E7:48_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E10:48_3"/><i><font color="Green">E45</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:48_3"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E12:48_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E5:48_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ordinal3.html#E9:48_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:48_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E14:48_3"/><b>then </b><i><font color="Green">E46</font></i>: 
(  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E6:48_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal3.html#T17" target="_self">Th17</a>, <a class="ref" href="ordinal2.html#T35">ORDINAL2:35</a></i>;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_2"/>
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> = 
 <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:48_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal3.html#E3:48_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal3.html#E6:48_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal3.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:48_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal3.html#E5:48_3"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="ordinal3.html#T24" target="_self">Th24</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="ordinal1.html#D6">ORDINAL1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="ordinal3.html#T26" target="_self">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:48_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal3.html#E5:48_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ordinal2.html#T54">ORDINAL2:54</a></i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_3"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:48_3_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a></i>;<br/><b>hence </b><a NAME="E3:48_3_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:48_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:48"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="ref" href="ordinal3.html#T29" target="_self">Th29</a>, <a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, <a class="txt" href="ordinal3.html#E3:48"><i><font color="Green">E35</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:48"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/>


</div>
