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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E4</font></i>: 
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) ) ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</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>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D13">ORDINAL2:def 13</a><br/>

<a NAME="E2:5"/><i><font color="Green">E5</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1"><b>case </b></a><a NAME="E1:5_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:5_1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:5"><i><font color="Green">E4</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E4:5_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="ordinal4.html#E3:5_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E5:5_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><b>then </b><i><font color="Green">E8</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#D6">ORDINAL3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T33">ORDINAL3:33</a></i>
;<br/><a NAME="E7:5_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="ordinal4.html#E5:5_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="ordinal3.html#T25">ORDINAL3:25</a>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><a NAME="E8:5_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:5_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="ordinal4.html#E6:5_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="ordinal4.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E9:5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2"><b>case </b></a><a NAME="E1:5_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E2:5_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:5_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:5_1_2"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:5"><i><font color="Green">E4</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E6:5_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="ordinal4.html#E4:5_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E7:5_1_2"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><b>then </b><i><font color="Green">E9</font></i>: <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#D6">ORDINAL3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T33">ORDINAL3:33</a></i>
;<br/><a NAME="E9:5_1_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="ordinal4.html#E7:5_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="ordinal3.html#T25">ORDINAL3:25</a>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><a NAME="E10:5_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:5_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="ordinal4.html#E8:5_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="ordinal4.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E11:5_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal4.html#K1" target="_self">^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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