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<span class="kw">let </span><font color="Maroon" title="c1">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st <font color="Olive" title="b1">f</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> &amp;  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Olive" title="b1">f</font> ) ) holds <br/> <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</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="c2">f</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp;  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  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/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> ) ) implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E27">Z0</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a> 
 <span class="kw">and </span><br/><a NAME="E2:59"/><span class="lab"><font color="Green" title="E28">Z1</font></span>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp;  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E3:59"/><span class="lab"><font color="Green" title="E29">Z2</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/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">l</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E28">Z1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_1"/>
(  ex <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 <br/>( <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> &amp; (  for <font color="Olive" title="b2">b</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="b2">b</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="b2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">a</font> ) ) or  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>   holds <br/>(  not <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> or  ex <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  not <font color="Olive" title="b2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">a</font> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_1_1"/>
 ex <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 <br/>( <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> &amp; (  for <font color="Olive" title="b2">b</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="b2">b</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="b2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">a</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E3:59_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; (  for <font color="Olive" title="b1">b</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">b</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">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">a</font> ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_1_1"/>
<font color="Maroon" title="c5">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E29">Z2</font></span></a></span>;<br/><a NAME="E4:59_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/><a NAME="E5:59_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:59_1_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:59_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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/></div><span class="kw">end;</span></div><a NAME="E5:59_1_1"/><span class="kw">then </span>
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a>, <a class="ref" href="zfmisc_1.html#T76" title="ZFMISC_1:th.76">ZFMISC_1:76</a></span>;<br/><a NAME="E6:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:59_1_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E29">Z2</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/><a NAME="E9:59_1_1"/>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/><a NAME="E10:59_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E8:59_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/><a NAME="E11:59_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E12:59_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_1_2"/><span class="lab"><font color="Green" title="E30">A3</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>   holds <br/>(  not <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> or  ex <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  not <font color="Olive" title="b2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">a</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">y0</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_2"/><span class="lab"><font color="Green" title="E31">B0</font></span>: 
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">x0</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E28">Z1</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E29">Z2</font></span></a></span>;<br/><a NAME="E4:59_1_2"/><span class="lab"><font color="Green" title="E32">B1</font></span>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_2"><span class="lab"><font color="Green" title="E31">B0</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E6:59_1_2"/><span class="lab"><font color="Green" title="E33">B2</font></span>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">b</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c5">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_2"><span class="lab"><font color="Green" title="E31">B0</font></span></a>, <a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/><a NAME="E7:59_1_2"/><span class="lab"><font color="Green" title="E34">C3</font></span>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1_2"><span class="lab"><font color="Green" title="E32">B1</font></span></a>, <a class="txt" href="#E6:59_1_2"><span class="lab"><font color="Green" title="E33">B2</font></span></a>, <a class="ref" href="ordinal6.html#T4" target="_self" title="ORDINAL6:th.4">EXCH1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">l</font> implies  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">l</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_2_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c8">x</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="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:59_1_2_1"/><span class="lab"><font color="Green" title="E36">B4</font></span>: 
( <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c9">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E29">Z2</font></span></a>, <a class="txt" href="#E3:59_1_2_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a></span>;<br/><a NAME="E6:59_1_2_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1_2_1"><span class="lab"><font color="Green" title="E36">B4</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E8:59_1_2_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c10">b</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c9">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:59_1_2_1"><span class="lab"><font color="Green" title="E36">B4</font></span></a></span>;<br/><a NAME="E9:59_1_2_1"/>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_2_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E5:59_1_2_1"><span class="lab"><font color="Green" title="E36">B4</font></span></a>, <a class="txt" href="#E6:59_1_2_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E8:59_1_2_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="ordinal6.html#T4" target="_self" title="ORDINAL6:th.4">EXCH1</a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E10:59_1_2_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E11:59_1_2_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_1_2_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">l</font> = <font color="Maroon" title="c3">l</font> as   <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_1_2"><span class="lab"><font color="Green" title="E34">C3</font></span></a>, <a class="txt" href="#E6:59_1_2"><span class="lab"><font color="Green" title="E33">B2</font></span></a>, <a class="ref" href="ordinal1.html#T28" title="ORDINAL1:th.28">ORDINAL1:28</a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">thus </span><a NAME="E10:59_1_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E28">Z1</font></span></a>, <a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E27">Z0</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">l</font> as   <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E27">Z0</font></span></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></span>;<br/><a NAME="E12:59_1_2"/>
<font color="Maroon" title="c7">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E28">Z1</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E13:59_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><a NAME="E14:59_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c8">g</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T6" title="ORDINAL5:th.6">ORDINAL5:6</a></span>;<br/><a NAME="E15:59_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">l</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E27">Z0</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E28">Z1</font></span></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></span>;<br/><a NAME="E16:59_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c8">g</font> &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c8">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="ordinal2.html#D10" title="ORDINAL2:def.10">ORDINAL2:def 10</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> )</span><br/><span class="kw">assume </span><a NAME="E17:59_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ; <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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:59_1_2"/><span class="lab"><font color="Green" title="E38">C1</font></span>: 
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59_1_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E21:59_1_2"/><span class="lab"><font color="Green" title="E39">C2</font></span>: 
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c11">y</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="#E13:59_1_2"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E19:59_1_2"><span class="lab"><font color="Green" title="E38">C1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E23:59_1_2"/><span class="lab"><font color="Green" title="E40">C4</font></span>: 
( <font color="Maroon" title="c11">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c12">t</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:59_1_2"><span class="lab"><font color="Green" title="E39">C2</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E29">Z2</font></span></a></span>;<br/><a NAME="E24:59_1_2"/><span class="lab"><font color="Green" title="E41">C3</font></span>: 
( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:59_1_2"><span class="lab"><font color="Green" title="E40">C4</font></span></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">z</font> = <font color="Maroon" title="c10">z</font>, <font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c12">t</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:59_1_2"><span class="lab"><font color="Green" title="E38">C1</font></span></a></span>;<br/><a NAME="E26:59_1_2"/>
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:59_1_2"><span class="lab"><font color="Green" title="E38">C1</font></span></a>, <a class="txt" href="#E21:59_1_2"><span class="lab"><font color="Green" title="E39">C2</font></span></a>, <a class="txt" href="#E23:59_1_2"><span class="lab"><font color="Green" title="E40">C4</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E27:59_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E27">Z0</font></span></a>, <a class="txt" href="#E24:59_1_2"><span class="lab"><font color="Green" title="E41">C3</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/><a NAME="E28:59_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:59_1_2"><span class="lab"><font color="Green" title="E38">C1</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E29:59_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:59_1_2"><span class="lab"><font color="Green" title="E40">C4</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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