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<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<a NAME="E1:35"/>
( ( <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">B</font> implies ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> ) ) &amp; ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> implies <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">B</font> ) &amp; (  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> implies (  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> &amp;  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> ) ) &amp; (  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> &amp;  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> implies  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E2:35"/><b>then </b>
( ( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> ) &amp; (  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) &amp; (  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> implies <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) &amp; (  not  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> or  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> or  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> ) &amp; ( (  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> or  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> ) implies  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> ) &amp; (  not <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) &amp; ( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) implies <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> ) )
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:35"/>
( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> iff  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3" target="_self">alef</a> <font color="Maroon">B</font> )
 ;<br/>


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