<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">D1</font>, <font color="Olive" title="b3">C2</font>, <font color="Olive" title="b4">D2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b4">D2</font> &amp; <font color="Olive" title="b2">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b4">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( <font color="Olive" title="b1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">C2</font> &amp; <font color="Olive" title="b2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">D2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">D1</font>, <font color="Maroon" title="c4">C2</font>, <font color="Maroon" title="c5">D2</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font> &amp; <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies ( <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> &amp; <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font>
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c5">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> &amp; <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font>;<br/>
<div><span class="lab"><font color="Green" title="E38">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1"/>
<font color="Maroon" title="c4">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:48_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">C</font>
 <span class="kw">and </span><br/><a NAME="E4:48_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c7">C</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="ref" href="ordinal3.html#T28" target="_self" title="ORDINAL3:th.28">Th31</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="ordinal3.html#T46" target="_self" title="ORDINAL3:th.46">Th54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T30" target="_self" title="ORDINAL3:th.30">Th33</a></span>
;<br/><a NAME="E6:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c5">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="ordinal3.html#T21" target="_self" title="ORDINAL3:th.21">Th24</a></span>;<br/><a NAME="E7:48_1"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T24" target="_self" title="ORDINAL3:th.24">Th27</a></span>;<br/><a NAME="E8:48_1"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="ordinal3.html#T36" target="_self" title="ORDINAL3:th.36">Th39</a></span>;<br/><span class="kw">hence </span><a NAME="E9:48_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E7:48_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="48_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_2"/>
<font color="Maroon" title="c2">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:48_2"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">C</font>
 <span class="kw">and </span><br/><a NAME="E4:48_2"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon" title="c7">C</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="ref" href="ordinal3.html#T28" target="_self" title="ORDINAL3:th.28">Th31</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="ordinal3.html#T46" target="_self" title="ORDINAL3:th.46">Th54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T30" target="_self" title="ORDINAL3:th.30">Th33</a></span>
;<br/><a NAME="E6:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T21" target="_self" title="ORDINAL3:th.21">Th24</a></span>;<br/><a NAME="E7:48_2"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T24" target="_self" title="ORDINAL3:th.24">Th27</a></span>;<br/><a NAME="E8:48_2"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="ordinal3.html#T36" target="_self" title="ORDINAL3:th.36">Th39</a></span>;<br/><span class="kw">hence </span><a NAME="E9:48_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E6:48_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:48"/>
<font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="ordinal1.html#T14" title="ORDINAL1:th.14">ORDINAL1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font></span><br/>

<span class="kw">hence </span><a NAME="E7:48"/>
<font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="ordinal3.html#T21" target="_self" title="ORDINAL3:th.21">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
