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<span class="kw">let </span><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   <a href="zf_refle.html#NM2" title="ZF_REFLE:NM.2">DOMAIN-Sequence</a> of <font color="Maroon" title="c1">W</font>  st (  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">b</font> ) &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds <br/> ( <font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Olive" title="b1">L</font> &amp; <font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  ) ) &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Olive" title="b1">L</font> is  <a href="zf_fund2.html#V1" title="ZF_FUND2:attr.1">predicatively_closed</a>  holds <br/> <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Olive" title="b1">L</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K10" title="ZF_MODEL:func.10">the_axiom_of_power_sets</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="zf_refle.html#NM2" title="ZF_REFLE:NM.2">DOMAIN-Sequence</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds <br/> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  ) ) &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> is  <a href="zf_fund2.html#V1" title="ZF_FUND2:attr.1">predicatively_closed</a>  implies  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K10" title="ZF_MODEL:func.10">the_axiom_of_power_sets</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:5"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds <br/> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  )
 <span class="kw">and </span><br/><a NAME="E3:5"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> is  <a href="zf_fund2.html#V1" title="ZF_FUND2:attr.1">predicatively_closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K10" title="ZF_MODEL:func.10">the_axiom_of_power_sets</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>;<br/>
<a NAME="E4:5"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c6">f</font> =  the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:5_1"/>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c4">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font> ;<br/>
<a NAME="E3:5_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<a NAME="E4:5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="zf_fund2.html#K1" title="ZF_FUND2:func.1">Section</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="zf_lang.html#K10" title="ZF_LANG:func.10">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<font color="Maroon" title="c7">e</font>)</span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>;<br/>
<a NAME="E5:5_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="zf_fund2.html#T1" target="_self" onmouseover="rs('zf_fund2/T1')" onmouseout="rh()">Th1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="lab"><font color="Green" title="E9">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> holds <br/><span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2;<br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> implies <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> )</span><br/><span class="kw">assume </span><a NAME="E1:5_2"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span><br/><a NAME="E2:5_2"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D1" onmouseover="rs('zfmisc_1/D1')" onmouseout="rh()">ZFMISC_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">D</font> = <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:5_2"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E2:5_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:5_2"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font>  ex <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">a</font>]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Olive" title="b1">a</font>]</span><br/>

<a NAME="E1:5_2_1"/>
<font color="Maroon" title="c6">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">u2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:5_2_1"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<font color="Maroon" title="c7">u2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">and </span><br/><a NAME="E4:5_2_1"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="zf_fund2.html#E4"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<a NAME="E5:5_2_1"/>
<font color="Maroon" title="c7">u2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_2_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="zf_refle.html#D2" onmouseover="rs('zf_refle/D2')" onmouseout="rh()">ZF_REFLE:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">u2</font> = <font color="Maroon" title="c7">u2</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T7" onmouseover="rs('zf_refle/T7')" onmouseout="rh()">ZF_REFLE:7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">u2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c8">u2</font>]</span><br/>

<span class="kw">thus </span><a NAME="E7:5_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c8">u2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_2_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">consider </span><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:5_2"/><span class="lab"><font color="Green" title="E14">A13</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font> &amp; (  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font>  ex <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">a</font>] &amp; (  for <font color="Olive" title="b3">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b3">b</font>] holds <br/><font color="Olive" title="b2">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b3">b</font> ) ) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S1" onmouseover="rs('zf_refle/S1')" onmouseout="rh()">ZF_REFLE:sch 1</a>(<a class="txt" href="#E5:5_2"><span class="lab"><font color="Green" title="E13">A10</font></span></a>);<br/></span><a NAME="E8:5_2"/><span class="lab"><font color="Green" title="E15">A14</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> or <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_2_2"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">u1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:5_2_2"/><span class="lab"><font color="Green" title="E16">A15</font></span>: 
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">and </span><br/><a NAME="E4:5_2_2"/><span class="lab"><font color="Green" title="E17">A16</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">u1</font> = <font color="Maroon" title="c8">u1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_2_2"><span class="lab"><font color="Green" title="E16">A15</font></span></a></span>;<br/>
<a NAME="E6:5_2_2"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u1</font> &amp; <font color="Maroon" title="c9">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> &amp; (  for <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c9">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">b</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:5_2_2"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_2_2"><span class="lab"><font color="Green" title="E17">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E9:5_2"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E10:5_2"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="classes2.html#T59" onmouseover="rs('classes2/T59')" onmouseout="rh()">CLASSES2:59</a></span>;<br/><a NAME="E11:5_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="classes1.html#D1" onmouseover="rs('classes1/D1')" onmouseout="rh()">CLASSES1:def 1</a></span>;<br/><a NAME="E12:5_2"/><span class="kw">then </span>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T1" onmouseover="rs('classes2/T1')" onmouseout="rh()">CLASSES2:1</a>, <a class="ref" href="relat_1.html#T113" onmouseover="rs('relat_1/T113')" onmouseout="rh()">RELAT_1:113</a></span>;<br/><a NAME="E13:5_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="card_1.html#T67" onmouseover="rs('card_1/T67')" onmouseout="rh()">CARD_1:67</a>, <a class="ref" href="ordinal1.html#T12" onmouseover="rs('ordinal1/T12')" onmouseout="rh()">ORDINAL1:12</a></span>;<br/><a NAME="E14:5_2"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="classes1.html#T1" onmouseover="rs('classes1/T1')" onmouseout="rh()">CLASSES1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font></span>)</span> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T19" onmouseover="rs('zf_refle/T19')" onmouseout="rh()">ZF_REFLE:19</a></span>;<br/><a NAME="E16:5_2"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
<font color="Maroon" title="c5">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">u2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> or <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_2_3"/>
<font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c8">u2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">c</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E4:5_2_3"/><span class="lab"><font color="Green" title="E18">A19</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d</font>
 <span class="kw">and </span><br/><a NAME="E5:5_2_3"/><span class="lab"><font color="Green" title="E19">A20</font></span>: 
<font color="Maroon" title="c9">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">and </span><br/><a NAME="E6:5_2_3"/>
 for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c9">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c10">c</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a></span>;<br/>
<a NAME="E7:5_2_3"/>
<font color="Maroon" title="c10">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E4:5_2_3"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:5_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="ordinal2.html#T19" onmouseover="rs('ordinal2/T19')" onmouseout="rh()">ORDINAL2:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:5_2_3"/>
<font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_2_3"><span class="lab"><font color="Green" title="E19">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E17:5_2"/><span class="lab"><font color="Green" title="E18">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a>, <a class="ref" href="zf_refle.html#T16" onmouseover="rs('zf_refle/T16')" onmouseout="rh()">ZF_REFLE:16</a></span>;<br/><a NAME="E18:5_2"/>
<font color="Maroon" title="c5">D</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>;<br/><a NAME="E19:5_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5_2"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a></span>;<br/><a NAME="E20:5_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:5_2"><span class="lab"><font color="Green" title="E18">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E21:5_2"/>
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E4:5_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E16:5_2"><span class="lab"><font color="Green" title="E17">A18</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E6:5"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> or <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_3"/>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">u</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:5_3"/><span class="lab"><font color="Green" title="E10">A22</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">and </span><br/><a NAME="E4:5_3"/><span class="lab"><font color="Green" title="E11">A23</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="zf_fund2.html#E4"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">u</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_3"><span class="lab"><font color="Green" title="E10">A22</font></span></a></span>;<br/>
<a NAME="E6:5_3"/>
<font color="Maroon" title="c6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_3"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="ref" href="zf_refle.html#D2" onmouseover="rs('zf_refle/D2')" onmouseout="rh()">ZF_REFLE:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">u</font> = <font color="Maroon" title="c6">u</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T7" onmouseover="rs('zf_refle/T7')" onmouseout="rh()">ZF_REFLE:7</a></span>;<br/>
<a NAME="E8:5_3"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">u</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<a NAME="E9:5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A24</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_3"><span class="lab"><font color="Green" title="E11">A23</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">u1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> )</span><br/>

<a NAME="E10:5_3"/><span class="lab"><font color="Green" title="E13">A25</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T16" onmouseover="rs('zf_refle/T16')" onmouseout="rh()">ZF_REFLE:16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:5_3"/>
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span><br/>

<a NAME="E12:5_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5_3"><span class="lab"><font color="Green" title="E12">A24</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:5_3"/>
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:5_3"><span class="lab"><font color="Green" title="E13">A25</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E7:5"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K10" title="ZF_MODEL:func.10">the_axiom_of_power_sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="zfmodel1.html#T9" onmouseover="rs('zfmodel1/T9')" onmouseout="rh()">ZFMODEL1:9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
