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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="b2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">A</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Olive" title="b2">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
( ( <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>  &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="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( 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="c2">A</font> &amp; <font color="Maroon" title="c2">A</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" 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">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span><br/>

<a NAME="E2:7"/><span class="lab"><font color="Green" title="E8">A2</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1"/>
( <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <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>  or <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </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="7_1_1"><span class="kw">case </span></a><a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <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> 
 ; <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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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"><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="E3:7_1_1"/><span class="lab"><font color="Green" title="E10">A4</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="E4:7_1_1"/><span class="lab"><font color="Green" title="E11">A5</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="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="ordinal3.html#T29" title="ORDINAL3:th.29">ORDINAL3:29</a></span>;<br/><span class="kw">take </span>
<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="c4">A1</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c4">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> or  not <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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</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/><span class="kw">thus </span><a NAME="E5:7_1_1"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</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">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c4">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> or  not <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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</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/><span class="kw">let </span><font color="Maroon" title="c5">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"> (  not <font color="Maroon" title="c4">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">C</font> or  not <font color="Maroon" title="c5">C</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="E6:7_1_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">and </span><br/><a NAME="E7:7_1_1"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c5">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="E8:7_1_1"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</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="c5">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:7_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</a></span>;<br/><a NAME="E9:7_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="c5">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="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E4:7_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E6:7_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E7:7_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="#E1:7_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E8:7_1_1"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="ordinal3.html#T29" title="ORDINAL3:th.29">ORDINAL3:29</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="7_1_2"><span class="kw">case </span></a><a NAME="E1:7_1_2"/>
<font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_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="E1:7_1_2_1_1"/>
( <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>  or <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<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> 
 ; <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">B1</font>, <font color="Olive" title="b2">B2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B2</font> holds <br/> ex <font color="Olive" title="b3">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> &amp; ( for <font color="Olive" title="b4">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b3">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b4">C</font> &amp; <font color="Olive" title="b4">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> holds <br/>( <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B2</font> ) ) )</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="E3:7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E10">A10</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="E4:7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E11">A11</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="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c5">B1</font>, <font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B2</font> implies  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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> holds <br/>( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c6">B2</font> ) ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E6:7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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" 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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> holds <br/>( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c6">B2</font> ) ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c7">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="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> holds <br/>( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c6">B2</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E7:7_1_2_1_1_1"/>
<font color="Maroon" title="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</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="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> holds <br/>( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c6">B2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c7">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> implies ( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B2</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<font color="Maroon" title="c7">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font>
 <span class="kw">and </span><br/><a NAME="E9:7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E15">A15</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B2</font> )</span><br/><a NAME="E10:7_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</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="c8">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E9:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E11:7_1_2_1_1_1"/>
( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E4:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E5:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="txt" href="#E6:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="txt" href="#E8:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E9:7_1_2_1_1_1"><span class="lab"><font color="Green" title="E15">A15</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E9">A16</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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">B1</font>, <font color="Olive" title="b2">B2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B2</font> holds <br/> ex <font color="Olive" title="b3">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> &amp; ( for <font color="Olive" title="b4">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b3">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b4">C</font> &amp; <font color="Olive" title="b4">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> holds <br/>( <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B2</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"> ( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> implies  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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E10">A17</font></span>: 
<font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E3:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E11">A18</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</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" 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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="E4:7_1_2_1_1_2"/>
<font color="Maroon" title="c4">B1</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">B</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="#E1:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E9">A16</font></span></a>, <a class="txt" href="#E2:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E10">A17</font></span></a>, <a class="ref" href="ordinal3.html#T73" title="ORDINAL3:th.73">ORDINAL3:73</a></span>;<br/><span class="kw">then </span><span class="kw">consider </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><br/><a NAME="E6:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E12">A19</font></span>: 
<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> <font color="Maroon" title="c1">fi</font>
 <span class="kw">and </span><br/><a NAME="E7:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E13">A20</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="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> <font color="Maroon" title="c1">fi</font> holds <br/>( <font color="Maroon" title="c4">B1</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">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="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> <font color="Maroon" title="c5">B2</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E11">A18</font></span></a>, <a class="ref" href="ordinal3.html#T74" title="ORDINAL3:th.74">ORDINAL3:74</a></span>;<br/><a NAME="E8:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E14">A21</font></span>: 
<font color="Maroon" title="c4">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B1</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T75" title="ORDINAL3:th.75">ORDINAL3:75</a></span>;<br/><a NAME="E9:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E15">A22</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></span>;<br/><a NAME="E10:7_1_2_1_1_2"/>
<font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E11">A18</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E11:7_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E12:7_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A23</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B2</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D6" title="ORDINAL3:def.6">ORDINAL3:def 6</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c7">A1</font> = <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"> ( <font color="Maroon" title="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="E13:7_1_2_1_1_2"/>
<font color="Maroon" title="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E12">A19</font></span></a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</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="c7">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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c8">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="c7">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="E14:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E17">A24</font></span>: 
<font color="Maroon" title="c7">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">C</font>
 <span class="kw">and </span><br/><a NAME="E15:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E18">A25</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> )</span><br/><a NAME="E16:7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E19">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</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="c8">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E15:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E18">A25</font></span></a>, <a class="ref" href="ordinal3.html#D2" title="ORDINAL3:def.2">ORDINAL3:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">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="c8">C</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E18:7_1_2_1_1_2"/>
<font color="Maroon" title="c4">B1</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E13">A20</font></span></a>, <a class="txt" href="#E14:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A24</font></span></a>, <a class="txt" href="#E15:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E18">A25</font></span></a></span>;<br/><a NAME="E19:7_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A27</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B1</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T49" title="ORDINAL2:th.49">ORDINAL2:49</a></span>;<br/><a NAME="E20:7_1_2_1_1_2"/>
<font color="Maroon" title="c9">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E13">A20</font></span></a>, <a class="txt" href="#E14:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A24</font></span></a>, <a class="txt" href="#E15:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E18">A25</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E21:7_1_2_1_1_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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="#E8:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A21</font></span></a>, <a class="txt" href="#E16:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E19">A26</font></span></a>, <a class="txt" href="#E12:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A23</font></span></a>, <a class="txt" href="#E19:7_1_2_1_1_2"><span class="lab"><font color="Green" title="E20">A27</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T49" title="ORDINAL2:th.49">ORDINAL2:49</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="E3:7_1_2"/>
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="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c3">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">A</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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="c3">B</font> <a href="ordinal3.html#K1" title="ORDINAL3:func.1">+^</a> <font color="Maroon" title="c1">fi</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> ) ) ) ) )
 ; <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>

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