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<b>let </b><font color="Maroon" title="c1">fi</font> be   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c1">fi</font>
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E9"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>case </b></a><a NAME="E1:8_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:8_1_1"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1_1"/>
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font>;<br/><b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><b>take </b><font color="Maroon" title="c6">A1</font> = <font color="Maroon" title="c5">x</font>;<br/><b>thus </b><a NAME="E4:8_1_1_1_1_1"/>
<font color="Maroon" title="c6">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c7">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E5:8_1_1_1_1_1"/>
( <font color="Maroon" title="c6">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> )
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font>
<b>by </b><i><a class="txt" href="ordinal4.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:8_1_1_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:8_1_1_1_1_2"/><i><font color="Green" title="E11">A5</font></i>: 
( <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c4">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> holds <br/><font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="ordinal4.html#E2:8_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></i>;<br/><b>take </b><font color="Maroon" title="c5">A1</font> = <font color="Maroon" title="c4">A1</font>;<br/><b>thus </b><a NAME="E4:8_1_1_1_1_2"/>
<font color="Maroon" title="c5">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:8_1_1_1_1_2"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/><b>let </b><font color="Maroon" title="c6">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E5:8_1_1_1_1_2"/>
( <font color="Maroon" title="c5">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> )
 ;<br/><a NAME="E6:8_1_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="ordinal4.html#E3:8_1_1_1_1_2"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/><b>hence </b><a NAME="E7:8_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:8_1_1"/>
 ex <font color="Olive" title="b1">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> &amp; ( for <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>case </b></a><a NAME="E1:8_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:8_1_2"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</a></i>;<br/><b>let </b><font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c5">B2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E3:8_1_2"/><i><font color="Green" title="E11">A7</font></i>: 
( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> )
 ;<br/><div><i><font color="Green" title="E12">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c6">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E2:8_1_2_1"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">A1</font>
 ;<br/><a NAME="E3:8_1_2_1"/>
( <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8_1_2_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:8_1_2_1"/><i><font color="Green" title="E14">A10</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> holds <br/>( <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> ) ) )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></i>;<br/><b>take </b><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c7">D</font>;<br/><b>thus </b><a NAME="E6:8_1_2_1"/>
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:8_1_2_1"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/><b>let </b><font color="Maroon" title="c9">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E7:8_1_2_1"/><i><font color="Green" title="E15">A11</font></i>: 
( <font color="Maroon" title="c8">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">E</font> = <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">C</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E9:8_1_2_1"/><i><font color="Green" title="E16">A12</font></i>: 
( <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font> &amp; <font color="Maroon" title="c10">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="ordinal4.html#E5:8_1_2_1"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="ordinal4.html#E7:8_1_2_1"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/><a NAME="E10:8_1_2_1"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">E</font> &amp; <font color="Maroon" title="c10">E</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8_1_2_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/><b>hence </b><a NAME="E11:8_1_2_1"/>
( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:8_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="ordinal4.html#E9:8_1_2_1"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_2_2"/>
for <font color="Olive" title="b1">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A1</font>
 ;<br/><a NAME="E2:8_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:8_1_2_2"/><i><font color="Green" title="E13">A13</font></i>: 
( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:8_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="ordinal3.html#T49" title="ORDINAL3:th.49">ORDINAL3:49</a></i>;<br/><a NAME="E5:8_1_2_2"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:8_1_2_2"/><i><font color="Green" title="E14">A14</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> &amp; ( for <font color="Olive" title="b1">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">fi</font> holds <br/>( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> ) ) )
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="ordinal4.html#E4:8_1_2_2"><i><font color="Green" title="E13">A13</font></i></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></i>;<br/><b>take </b><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c7">D</font>;<br/><b>thus </b><a NAME="E8:8_1_2_2"/>
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E7:8_1_2_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/><b>let </b><font color="Maroon" title="c9">A1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E9:8_1_2_2"/><i><font color="Green" title="E15">A15</font></i>: 
( <font color="Maroon" title="c8">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">A1</font> &amp; <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">E</font> = <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A1</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E11:8_1_2_2"/>
( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A1</font> &amp; <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="ordinal4.html#E7:8_1_2_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="ordinal4.html#E9:8_1_2_2"><i><font color="Green" title="E15">A15</font></i></a></i>;<br/><a NAME="E12:8_1_2_2"/><b>then </b>
( <font color="Maroon" title="c6">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c10">E</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="ordinal2.html#T57" title="ORDINAL2:th.57">ORDINAL2:57</a></i>;<br/><a NAME="E13:8_1_2_2"/><b>then </b>
( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c10">E</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c10">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A1</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="ordinal4.html#E4:8_1_2_2"><i><font color="Green" title="E13">A13</font></i></a>, <a class="txt" href="ordinal4.html#E9:8_1_2_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal2.html#T58" title="ORDINAL2:th.58">ORDINAL2:58</a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/><b>hence </b><a NAME="E14:8_1_2_2"/>
( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:8_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:8_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> or ( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> ) ) ) )
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:8_1_2"><i><font color="Green" title="E12">A8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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