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<span class="kw">let </span><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A1</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c1">fi</font> &amp; <font color="Maroon" title="c4">A2</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c1">fi</font> implies <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
( ( <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</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="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</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> <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="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>  ) ) ) or ( <font color="Maroon" title="c3">A1</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; ( for <font color="Olive" title="b1">B</font>, <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">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font> &amp; <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> holds <br/> ex <font color="Olive" title="b3">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">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="b4">E</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b3">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b4">E</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="Olive" title="b1">B</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="b4">E</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="b4">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> ) ) ) ) ) )
 <span class="kw">and </span><br/><a NAME="E2:62_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
( ( <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</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="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</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> <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="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>  ) ) ) or ( <font color="Maroon" title="c4">A2</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; ( for <font color="Olive" title="b1">B</font>, <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">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A2</font> &amp; <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> holds <br/> ex <font color="Olive" title="b3">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">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="b4">E</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b3">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b4">E</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="Olive" title="b1">B</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="b4">E</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="b4">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> ) ) ) ) ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal2.html#D13" target="_self" title="ORDINAL2:def.13">ORDINAL2:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font></span><br/>

<div><span class="lab"><font color="Green" title="E17">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A1</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><span class="kw">assume </span><a NAME="E3:62_1_1_1"/>
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E5:62_1_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:62_1_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
for <font color="Olive" title="b1">A</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">A</font> &amp; <font color="Olive" title="b1">A</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="c3">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">A</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">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="c4">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_1_1_1_1"/>
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:62_1_1_1_1"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c8">B</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>
 <span class="kw">and </span><br/><a NAME="E4:62_1_1_1_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
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="c8">B</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/><a NAME="E5:62_1_1_1_1"/>
( <font color="Maroon" title="c8">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> or <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">B</font> )
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">E</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E7:62_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
( ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> ) or ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">B</font> ) )
 ;<br/><a NAME="E8:62_1_1_1_1"/>
<font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E3:62_1_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:62_1_1_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E7:62_1_1_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:62_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E6:62_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E3:62_1_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E7:62_1_1_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E9:62_1_1_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">and </span><br/><a NAME="E10:62_1_1_1"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c8">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</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">x</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">A</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">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="c3">A1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><a NAME="E11:62_1_1_1"/>
( <font color="Maroon" title="c8">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> or <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> )
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">E</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E13:62_1_1_1"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
( ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> ) or ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> ) )
 ;<br/><a NAME="E14:62_1_1_1"/>
<font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c3">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E9:62_1_1_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E10:62_1_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E13:62_1_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a></span>;<br/><a NAME="E15:62_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A13</font></span>: 
( <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font> or <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></span>;<br/><a NAME="E16:62_1_1_1"/>
<font color="Maroon" title="c3">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="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E6:62_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E9:62_1_1_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E13:62_1_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E17:62_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:62_1_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A2</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<span class="kw">assume </span><a NAME="E6:62_1_1"/>
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:62_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A2</font> or <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E9:62_1_1"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:62_1_1"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
for <font color="Olive" title="b1">A</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">A</font> &amp; <font color="Olive" title="b1">A</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="c4">A2</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">A</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">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="c3">A1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:62_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_1_1_2"/>
<font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:62_1_1_2"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c8">B</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>
 <span class="kw">and </span><br/><a NAME="E4:62_1_1_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
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="c8">B</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/><a NAME="E5:62_1_1_2"/>
( <font color="Maroon" title="c8">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> or <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">B</font> )
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">E</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E7:62_1_1_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
( ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> ) or ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">B</font> ) )
 ;<br/><a NAME="E8:62_1_1_2"/>
<font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_1_1"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E3:62_1_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E4:62_1_1_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:62_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_1_1"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E10:62_1_1"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E3:62_1_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E13:62_1_1"/><span class="lab"><font color="Green" title="E20">A19</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">and </span><br/><a NAME="E14:62_1_1"/><span class="lab"><font color="Green" title="E21">A20</font></span>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c8">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</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">x</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">A</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">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="c4">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E15:62_1_1"/>
( <font color="Maroon" title="c8">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> or <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">E</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E17:62_1_1"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
( ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> ) or ( <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> ) )
 ;<br/>
<a NAME="E18:62_1_1"/>
<font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c4">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_1_1"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E13:62_1_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E14:62_1_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E17:62_1_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a></span>;<br/>
<a NAME="E19:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A22</font></span>: 
( <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A2</font> or <font color="Maroon" title="c1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></span>;<br/>
<a NAME="E20:62_1_1"/>
<font color="Maroon" title="c4">A2</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">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_1_1"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E10:62_1_1"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E13:62_1_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E17:62_1_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:62_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:62_1_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
