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

<b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c4">M</font>, <font color="Maroon" title="c5">N</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>;<br/>







<a NAME="E1:76"/><i><font color="Green" title="E54">A1</font></i>: 
for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Olive" title="b1">K</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Olive" title="b2">L</font> &amp; <font color="Olive" title="b3">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Olive" title="b4">N</font> &amp; <font color="Olive" title="b2">L</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Olive" title="b4">N</font> holds <br/><font color="Olive" title="b1">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Olive" title="b3">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Olive" title="b2">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Olive" title="b4">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">K</font>, <font color="Maroon" title="c7">L</font>, <font color="Maroon" title="c8">M</font>, <font color="Maroon" title="c9">N</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>;<br/>







<b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E55">A2</font></i>: 
( <font color="Maroon" title="c6">K</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c7">L</font> &amp; <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c9">N</font> &amp; <font color="Maroon" title="c7">L</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c9">N</font> )
 ;<br/>

<a NAME="E2:76_1"/><b>then </b><i><font color="Green" title="E56">A3</font></i>: 
0 <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c7">L</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<div><i><font color="Green" title="E57">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c9">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">N</font> = <font color="Maroon" title="c9">N</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> ;<br/><b>reconsider </b><font color="Maroon" title="c11">L</font> = <font color="Maroon" title="c7">L</font>, <font color="Maroon" title="c12">M</font> = <font color="Maroon" title="c8">M</font>, <font color="Maroon" title="c13">K</font> = <font color="Maroon" title="c6">K</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_4.html#E1:76_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="card_4.html#E1:76_1_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="card_4.html#T26" target="_self" title="CARD_4:th.26">Th26</a></i>;<br/><a NAME="E4:76_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">K</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">M</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">N</font> )
 <b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/><a NAME="E5:76_1_1"/><b>then </b><i><font color="Green" title="E59">A6</font></i>: 
( <font color="Maroon" title="c13">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">M</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c10">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">L</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">N</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">L</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">N</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">M</font> )
 <b>by </b><i><a class="txt" href="card_4.html#E1:76_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="card_1.html#T73" title="CARD_1:th.73">CARD_1:73</a>, <a class="ref" href="card_2.html#T52" title="CARD_2:th.52">CARD_2:52</a></i>;<br/><a NAME="E6:76_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">N</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">N</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">L</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">N</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E7:76_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">L</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:76_1_1"/>
<font color="Maroon" title="c6">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c7">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c9">N</font>
 <b>by </b><i><a class="txt" href="card_4.html#E5:76_1_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="card_1.html#T73" title="CARD_1:th.73">CARD_1:73</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_2"/>
not <font color="Maroon" title="c9">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><a NAME="E2:76_1_2"/><b>then </b><i><font color="Green" title="E58">A7</font></i>: 
( <font color="Maroon" title="c7">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c9">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">N</font> &amp;  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">N</font> )
 <b>by </b><i><a class="txt" href="card_4.html#E1:76_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="card_4.html#E2:76_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self" title="CARD_4:th.11">Th11</a>, <a class="ref" href="card_4.html#T78" target="_self" title="CARD_4:th.78">Th78</a>, <a class="ref" href="card_1.html#T83" title="CARD_1:th.83">CARD_1:83</a></i>;<br/><a NAME="E3:76_1_2"/>
<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c8">M</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T25" title="ORDINAL1:th.25">ORDINAL1:25</a></i>;<br/><a NAME="E4:76_1_2"/><b>then </b>
( ( <font color="Maroon" title="c6">K</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c8">M</font> &amp; ( <font color="Maroon" title="c8">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  not <font color="Maroon" title="c8">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) ) or ( <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c6">K</font> &amp; ( <font color="Maroon" title="c6">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  not <font color="Maroon" title="c6">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/><a NAME="E5:76_1_2"/><b>then </b><i><font color="Green" title="E59">A8</font></i>: 
( ( <font color="Maroon" title="c6">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c8">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) or <font color="Maroon" title="c6">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c8">M</font> or ( <font color="Maroon" title="c6">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c6">K</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c9">N</font> ) )
 <b>by </b><i><a class="txt" href="card_4.html#E1:76_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="card_4.html#T26" target="_self" title="CARD_4:th.26">Th26</a>, <a class="ref" href="card_4.html#T79" target="_self" title="CARD_4:th.79">Th79</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_2_1"/>
( <font color="Maroon" title="c6">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c8">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">K</font> = <font color="Maroon" title="c6">K</font>, <font color="Maroon" title="c11">M</font> = <font color="Maroon" title="c8">M</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> ;<br/><a NAME="E3:76_1_2_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">K</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">M</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">K</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M</font> )
 <b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/><b>then </b><font color="Maroon" title="c10">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c11">M</font> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">K</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T52" title="CARD_2:th.52">CARD_2:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">K</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">M</font></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:76_1_2_1"/>
<font color="Maroon" title="c6">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c7">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c9">N</font>
 <b>by </b><i><a class="txt" href="card_4.html#E2:76_1_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:76_1_2"/>
<font color="Maroon" title="c6">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c7">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c9">N</font>
 <b>by </b><i><a class="txt" href="card_4.html#E1:76_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="card_4.html#E2:76_1_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="card_4.html#E5:76_1_2"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:76_1"/>
<font color="Maroon" title="c6">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c7">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c9">N</font>
 <b>by </b><i><a class="txt" href="card_4.html#E3:76_1"><i><font color="Green" title="E57">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:76"/>
<font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c5">N</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#T25" title="ORDINAL1:th.25">ORDINAL1:25</a></i>;<br/>
<a NAME="E3:76"/><b>then </b>
( <font color="Maroon" title="c2">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c3">L</font> &amp; ( <font color="Maroon" title="c3">L</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c5">N</font> or <font color="Maroon" title="c5">N</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c3">L</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:76"/>
( <font color="Maroon" title="c2">K</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c4">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c5">N</font> implies ( <font color="Maroon" title="c2">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c4">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c3">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c5">N</font> &amp; <font color="Maroon" title="c4">M</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c2">K</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c3">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c5">N</font> ) )
 <b>by </b><i><a class="txt" href="card_4.html#E1:76"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>


</div>
