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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> <a href="ordinal5.html#K2" title="ORDINAL5:func.2">first_epsilon_greater_than</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69"/><span class="lab"><font color="Green" title="E48">Z0</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <span class="kw">and </span><br/><a NAME="E2:69"/><span class="lab"><font color="Green" title="E49">Z1</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> <a href="ordinal5.html#K2" title="ORDINAL5:func.2">first_epsilon_greater_than</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies ( <font color="Maroon" title="c2">x</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp;  ex <font color="Olive" title="b1">e</font> being  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E50">Z2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</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="c2">x</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp;  ex <font color="Olive" title="b1">e</font> being  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) )</span><br/><span class="kw">hence </span><a NAME="E2:69_1"/>
<font color="Maroon" title="c2">x</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E48">Z0</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">  ex <font color="Olive" title="b1">e</font> being  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c4">e</font> =  <a href="ordinal5.html#K2" title="ORDINAL5:func.2">first_epsilon_greater_than</a> <font color="Maroon" title="c3">a</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/><span class="kw">thus </span><a NAME="E4:69_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E49">Z1</font></span></a>, <a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E50">Z2</font></span></a>, <a class="ref" href="ordinal5.html#D6" target="_self" title="ORDINAL5:def.6">FEGT</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="E4:69"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> is   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T49" target="_self" title="ORDINAL5:th.49">Th37</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/>


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