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<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon" title="c1">S</font> is <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a> )</span><br/>

<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green" title="E5">Z</font></i>: 
<font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a></span><br/>

<b>take </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D1" target="_self" title="CARD_1:def.1">CARD_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  holds <br/> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) )</span><br/>

<b>thus </b><a NAME="E2:10_1_1"/>
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:10_1_1"><i><font color="Green" title="E5">Z</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  holds <br/> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A</font></span><br/>

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  implies  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E3:10_1_1"/>
<font color="Maroon" title="c2">A</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>thus </b><a NAME="E4:10_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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