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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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"> ( <font color="Maroon" title="c1">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">y</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  &amp;  not <font color="Maroon" title="c1">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a> <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="c1">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:78"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><br/>

<a NAME="E3:78"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</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="ordinal2.html#T38" title="ORDINAL2:th.38">ORDINAL2:38</a></span>;<br/>
<a NAME="E4:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T36" target="_self" title="ORDINAL3:th.36">Th39</a></span>;<br/>
<a NAME="E5:78"/>
<font color="Maroon" title="c1">x</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="txt" href="#E2:78"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="ordinal2.html#T35" title="ORDINAL2:th.35">ORDINAL2:35</a></span>;<br/>
<a NAME="E6:78"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T36" target="_self" title="ORDINAL3:th.36">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:78"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</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/>


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