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<span class="kw">let </span><font color="Maroon" title="c1">fi</font> be   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c1">fi</font> holds <br/><font color="Olive" title="b1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Olive" title="b2">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c1">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><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>; <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="c2">A</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="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <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><br/>





<a NAME="E1:9"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:9"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<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>
 ; <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="c2">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <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><br/>

<a NAME="E3:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ordinal4.html#E9"><span class="lab"><font color="Green" title="E7">Lm4</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_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/> <span class="comment"><font color="Red">:: according to </font></span><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="9_1_1"><span class="kw">case </span></a><a NAME="E1:9_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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 <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#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) )</span><br/><div class="add"><a NAME="E2:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A4</font></span>: 
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_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="9_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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>  ) )</span><br/><div class="add"><span class="kw">consider </span><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/><span class="kw">reconsider </span><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/><span class="kw">take </span><font color="Maroon" title="c6">A1</font> = <font color="Maroon" title="c5">x</font>; <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="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> &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="c6">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> <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="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><br/><span class="kw">thus </span><a NAME="E4:9_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E11">A3</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"> 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="c6">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> <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="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><br/><span class="kw">let </span><font color="Maroon" title="c7">C</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="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> implies <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:9_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>
 <span class="kw">and </span><br/><a NAME="E6:9_1_1_1_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E6:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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>  ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:9_1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:9_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">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="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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E2:9_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">A1</font> = <font color="Maroon" title="c4">A1</font>; <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="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> &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="c5">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> <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="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><br/><span class="kw">thus </span><a NAME="E5:9_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="c5">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> <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="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><br/><span class="kw">let </span><font color="Maroon" title="c6">C</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="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> implies <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>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:9_1_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:9_1_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><a NAME="E8:9_1_1_1_1_2"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E7:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E9:9_1_1_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="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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E6:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E7:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:9_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:9_1_1"/>
 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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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 <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#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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2"><span class="kw">case </span></a><a NAME="E1:9_1_2"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <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="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> or  not <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="Olive">b<sub>2</sub></font> or  ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</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>4</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</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="Olive">b<sub>1</sub></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>4</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>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) )</span><br/><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or  not <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> or  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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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> ) ) ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:9_1_2"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:9_1_2"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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> ) ) ) )</span><br/><a NAME="E4:9_1_2"/><span class="lab"><font color="Green" title="E15">A15</font></span>: 
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_2"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</a></span>;<br/><div><span class="lab"><font color="Green" title="E16">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:9_1_2_1"/>
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">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> &amp; ( for <font color="Olive" title="b2">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">A1</font> &amp; <font color="Olive" title="b2">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> holds <br/>( <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" title="b2">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="Olive" title="b2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> ) ) )</span><br/><a NAME="E2:9_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_2_1"/><span class="lab"><font color="Green" title="E17">A17</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:9_1_2_1"/><span class="lab"><font color="Green" title="E18">A18</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_2"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="ordinal3.html#T49" title="ORDINAL3:th.49">ORDINAL3:49</a></span>;<br/><a NAME="E6:9_1_2_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</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="E8:9_1_2_1"/><span class="lab"><font color="Green" title="E19">A19</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="E9:9_1_2_1"/><span class="lab"><font color="Green" title="E20">A20</font></span>: 
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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:9_1_2_1"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c7">D</font>; <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="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> &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="c8">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> <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/>( <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" title="b1">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="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E10:9_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9_1_2_1"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="c8">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> <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/>( <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" title="b1">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="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">A1</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="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> implies ( <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> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:9_1_2_1"/><span class="lab"><font color="Green" title="E21">A21</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E12:9_1_2_1"/><span class="lab"><font color="Green" title="E22">A22</font></span>: 
<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>
 ; <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="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> )</span><br/><span class="kw">reconsider </span><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="E14:9_1_2_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">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E9:9_1_2_1"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="txt" href="#E11:9_1_2_1"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E12:9_1_2_1"><span class="lab"><font color="Green" title="E22">A22</font></span></a></span>;<br/><a NAME="E15:9_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/><a NAME="E16:9_1_2_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E9:9_1_2_1"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="txt" href="#E11:9_1_2_1"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E12:9_1_2_1"><span class="lab"><font color="Green" title="E22">A22</font></span></a></span>;<br/><a NAME="E17:9_1_2_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_2"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="ref" href="ordinal2.html#T57" title="ORDINAL2:th.57">ORDINAL2:57</a></span>;<br/><a NAME="E18:9_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_2_1"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/><a NAME="E19:9_1_2_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E12:9_1_2_1"><span class="lab"><font color="Green" title="E22">A22</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E20:9_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">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_2"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E15:9_1_2_1"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E18:9_1_2_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T58" title="ORDINAL2:th.58">ORDINAL2:58</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:9_1_2_2"/><span class="lab"><font color="Green" title="E17">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c6">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><a NAME="E3:9_1_2_2"/><span class="lab"><font color="Green" title="E18">A26</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">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> &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">D</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/>( <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" title="b2">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="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> ) ) )</span><br/><a NAME="E4:9_1_2_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_2_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</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="E6:9_1_2_2"/><span class="lab"><font color="Green" title="E19">A27</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="E7:9_1_2_2"/><span class="lab"><font color="Green" title="E20">A28</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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E1:9_1_2_2"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c7">D</font>; <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="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> &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="c8">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> <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/>( <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" title="b1">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="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E8:9_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_2_2"><span class="lab"><font color="Green" title="E19">A27</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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">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> <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/>( <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" title="b1">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="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">C</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="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> implies ( <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> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:9_1_2_2"/><span class="lab"><font color="Green" title="E21">A29</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:9_1_2_2"/><span class="lab"><font color="Green" title="E22">A30</font></span>: 
<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>
 ; <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="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> )</span><br/><span class="kw">reconsider </span><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="E12:9_1_2_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E7:9_1_2_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E9:9_1_2_2"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="txt" href="#E10:9_1_2_2"><span class="lab"><font color="Green" title="E22">A30</font></span></a></span>;<br/><a NAME="E13:9_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A31</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_2_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/><a NAME="E14:9_1_2_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E7:9_1_2_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E9:9_1_2_2"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="txt" href="#E10:9_1_2_2"><span class="lab"><font color="Green" title="E22">A30</font></span></a></span>;<br/><a NAME="E15:9_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A32</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/><a NAME="E16:9_1_2_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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E10:9_1_2_2"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E17:9_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">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_2"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="txt" href="#E3:9_1_2"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E13:9_1_2_2"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="txt" href="#E15:9_1_2_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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">hence </span><a NAME="E7:9_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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_2"><span class="lab"><font color="Green" title="E16">A16</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></div><span class="kw">end;</span></div>

</div>
