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<span class="kw">let </span><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">L</font>, <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</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="c1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> implies ( <font color="Maroon" title="c1">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c3">M</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c4">N</font> ) )</span><br/>







<a NAME="E1:84"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">L</font> &amp; <font color="Olive" title="b3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">N</font> &amp; <font color="Olive" title="b2">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="hidden.html#R2" title="HIDDEN:pred.2">in</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="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">K</font>, <font color="Maroon" title="c6">L</font>, <font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c8">N</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</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">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c6">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">N</font> implies <font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:84_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c5">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">and </span><br/><a NAME="E2:84_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">and </span><br/><a NAME="E3:84_1"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c6">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font></span><br/>

<div><span class="lab"><font color="Green" title="E19">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:84_1_1"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c8">N</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">N</font> = <font color="Maroon" title="c8">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/><span class="kw">reconsider </span><font color="Maroon" title="c10">L</font> = <font color="Maroon" title="c6">L</font>, <font color="Maroon" title="c11">M</font> = <font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c12">K</font> = <font color="Maroon" title="c5">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E2:84_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E3:84_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E1:84_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="card_3.html#T111" title="CARD_3:th.111">CARD_3:111</a></span>;<br/><a NAME="E4:84_1_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/><a NAME="E5:84_1_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/><a NAME="E6:84_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:84_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="nat_1.html#T42" title="NAT_1:th.42">NAT_1:42</a></span>;<br/><a NAME="E7:84_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">K</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">K</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E8:84_1_1"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/><a NAME="E9:84_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c10">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>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="card_2.html#T52" title="CARD_2:th.52">CARD_2:52</a></span>;<br/><a NAME="E10:84_1_1"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/><a NAME="E11:84_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">K</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E8:84_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="nat_1.html#T42" title="NAT_1:th.42">NAT_1:42</a></span>;<br/><a NAME="E12:84_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">K</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E13:84_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">K</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:84_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:84_1_1"/>
<font color="Maroon" title="c12">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">K</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">M</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:84_1_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E5:84_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="card_2.html#T52" title="CARD_2:th.52">CARD_2:52</a></span>;<br/><span class="kw">hence </span><a NAME="E15:84_1_1"/>
<font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:84_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E13:84_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="nat_1.html#T42" title="NAT_1:th.42">NAT_1:42</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>
<a NAME="E5:84_1"/><span class="lab"><font color="Green" title="E20">A14</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:84_1_2"/><span class="lab"><font color="Green" title="E21">A15</font></span>: 
not <font color="Maroon" title="c8">N</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font></span><br/><a NAME="E2:84_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A16</font></span>: 
<font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:84_1"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="card_4.html#T78" target="_self" title="CARD_4:th.78">Th78</a></span>;<br/><a NAME="E3:84_1_2"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
 <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="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_2"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="card_3.html#T102" title="CARD_3:th.102">CARD_3:102</a></span>;<br/><div><span class="lab"><font color="Green" title="E24">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:84_1_2_1"/>
( <font color="Maroon" title="c5">K</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c7">M</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">K</font> = <font color="Maroon" title="c5">K</font>, <font color="Maroon" title="c10">M</font> = <font color="Maroon" title="c7">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="E3:84_1_2_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">K</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c9">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c10">M</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">K</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T52" title="CARD_2:th.52">CARD_2:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">K</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:84_1_2_1"/>
<font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_2"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="txt" href="#E3:84_1_2"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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><a NAME="E5:84_1_2"/>
( ( <font color="Maroon" title="c5">K</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">M</font> &amp; ( <font color="Maroon" title="c7">M</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  or  not <font color="Maroon" title="c7">M</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) ) or ( <font color="Maroon" title="c7">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">K</font> &amp; ( <font color="Maroon" title="c5">K</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  or  not <font color="Maroon" title="c5">K</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) ) )
 ;<br/><a NAME="E6:84_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c5">K</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c7">M</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) or <font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">M</font> or ( <font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c5">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:84_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="card_4.html#T79" target="_self" title="CARD_4:th.79">Th79</a></span>;<br/><span class="kw">hence </span><a NAME="E7:84_1_2"/>
<font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:84_1_2"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="txt" href="#E4:84_1_2"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:84_1"/>
<font color="Maroon" title="c5">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1"><span class="lab"><font color="Green" title="E19">A5</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>
<a NAME="E2:84"/>
( <font color="Maroon" title="c2">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">N</font> or <font color="Maroon" title="c4">N</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">L</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:84"/>
( <font color="Maroon" title="c1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> implies ( <font color="Maroon" title="c1">K</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c3">M</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c4">N</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E15">A1</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>
