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<b>let </b><font color="Maroon">fi</font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green">E13</font></i>: 
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> holds <br/><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) ) ) or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/> ex <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">E</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> holds <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">E</font> &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) ) ) ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D13">ORDINAL2:def 13</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</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="7_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:7_1_1_2"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/><a NAME="E3:7_1_1_2"/><b>then </b>
 ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> holds <br/>(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> ) ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:7_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


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