<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN: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#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">a</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b2">a</font></span>)</span> ) &amp; ( for <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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="hidden.html#R2" title="HIDDEN: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_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> holds <br/> <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Olive" title="b1">L</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a></span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN: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#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">a</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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="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> &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_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> implies  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN: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>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green" title="E15">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">a</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span>
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green" title="E16">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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="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> &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> )
 <b>and </b><br/><a NAME="E5:13"/><i><font color="Green" title="E17">A5</font></i>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a></span><br/>

<a NAME="E6:13"/><i><font color="Green" title="E18">A6</font></i>: 
 <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" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">X</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> or <font color="Maroon" title="c3">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/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c3">X</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">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/>

<b>then consider </b><font color="Maroon" title="c4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c4">u</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">L</font> &amp; <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN: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="c4">u</font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">u</font> = <font color="Maroon" title="c4">u</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E3:13_1"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<a NAME="E5:13_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="#E3:13_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="zf_refle.html#D5" title="ZF_REFLE:def.5">ZF_REFLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">u</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E7:13_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">u</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>
 
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E8:13_1"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c3">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="c6">u</font>
 
<b>by </b><i><a class="txt" href="#E3:13_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">u1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c7">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/>

<b>assume </b><a NAME="E9:13_1"/>
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">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="E10:13_1"/><b>then </b>
( <font color="Maroon" title="c7">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="c6">u</font> &amp; <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">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> )
 
<b>by </b><i><a class="txt" href="#E8:13_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="zf_refle.html#T24" title="ZF_REFLE:th.24">ZF_REFLE:24</a></i>;<br/>
<b>hence </b><a NAME="E11:13_1"/>
<font color="Maroon" title="c7">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:13"/><b>then </b><i><font color="Green" title="E19">A9</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="zf_fund2.html#T5" target="_self" title="ZF_FUND2:th.5">Th5</a></i>;<br/>
<a NAME="E8:13"/>
for <font color="Olive" title="b1">u1</font>, <font color="Olive" title="b2">u2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b2">u2</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> holds <br/><span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">u2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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>
 
<b>by </b><i><a class="txt" href="#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="zf_fund1.html#T6" title="ZF_FUND1:th.6">ZF_FUND1:6</a></i>;<br/>
<a NAME="E9:13"/><b>then </b><i><font color="Green" title="E20">A10</font></i>: 
 <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#K7" title="ZF_MODEL:func.7">the_axiom_of_pairs</a> 
 
<b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="zfmodel1.html#T3" title="ZFMODEL1:th.3">ZFMODEL1:3</a></i>;<br/>
<a NAME="E10:13"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</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> holds <br/> <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">u</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>
 
<b>by </b><i><a class="txt" href="#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="zf_fund1.html#T2" title="ZF_FUND1:th.2">ZF_FUND1:2</a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green" title="E21">A11</font></i>: 
 <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#K8" title="ZF_MODEL:func.8">the_axiom_of_unions</a> 
 
<b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="zfmodel1.html#T5" title="ZFMODEL1:th.5">ZFMODEL1:5</a></i>;<br/>
<a NAME="E12:13"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</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> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b2">u1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/> ex <font color="Olive" title="b3">u2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b3">u2</font> &amp; <font color="Olive" title="b3">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> : <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 <a href="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c3">ksi</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:13_2"/><i><font color="Green" title="E22">A12</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">ksi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S11" title="NAT_1:sch.11">NAT_1:sch 11</a>();<br/></i>
<a NAME="E3:13_2"/><i><font color="Green" title="E23">A13</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> );<br/>
<a NAME="E1:13_2_1"/><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<div><i><font color="Green" title="E25">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:13_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> ;<br/><a NAME="E3:13_2_1_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="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>
 <b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:13_2_1_1"/><i><font color="Green" title="E26">A16</font></i>: 
( <font color="Maroon" title="c6">u</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">L</font> &amp; <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="hidden.html#R2" title="HIDDEN: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="c6">u</font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/><a NAME="E6:13_2_1_1"/><i><font color="Green" title="E27">A17</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="ref" href="zf_refle.html#D5" title="ZF_REFLE:def.5">ZF_REFLE:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">u</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E5:13_2_1_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/><a NAME="E8:13_2_1_1"/><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="#E5:13_2_1_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="#E6:13_2_1_1"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/><a NAME="E9:13_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> : <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="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E5:13_2_1_1"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/><a NAME="E10:13_2_1_1"/><b>then </b><i><font color="Green" title="E29">A19</font></i>: 
 <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> : <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="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> </span>}</span>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T22" title="ORDINAL2:th.22">ORDINAL2:22</a></i>;<br/><b>thus </b><a NAME="E11:13_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_1_1_1"/><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> : <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="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/>
<a NAME="E2:13_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="#E8:13_2_1_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="#E10:13_2_1_1"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1_1_1"/>
<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="#E1:13_2_1_1_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font></span><br/>

<b>hence </b><a NAME="E4:13_2_1_1_1"/>
<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:13_2_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E1:13_2_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="#E2:13_2_1"><i><font color="Green" title="E25">A15</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:13_2"/><i><font color="Green" title="E24">A21</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E1:13_2_2"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> : <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<a NAME="E3:13_2_2"/>
<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="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>
 
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:13_2_2"/><i><font color="Green" title="E26">A23</font></i>: 
( <font color="Maroon" title="c6">b</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">L</font> &amp; <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="hidden.html#R2" title="HIDDEN: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="c6">b</font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<a NAME="E6:13_2_2"/><i><font color="Green" title="E27">A24</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="#E5:13_2_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="zf_refle.html#D5" title="ZF_REFLE:def.5">ZF_REFLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E8:13_2_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="#E6:13_2_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E9:13_2_2"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> : <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="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E5:13_2_2"><i><font color="Green" title="E26">A23</font></i></a></i>;<br/>
<a NAME="E10:13_2_2"/><b>then </b>
<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> : <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E1:13_2_2"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="ordinal2.html#T25" title="ORDINAL2:th.25">ORDINAL2:25</a></i>;<br/>
<a NAME="E11:13_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <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="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">w</font> )
 
;<br/>
<b>hence </b><a NAME="E12:13_2_2"/>
<font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:13_2"/><i><font color="Green" title="E25">A25</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:13_2_3"/>
not <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:13_2_3"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/>
<a NAME="E5:13_2_3"/><b>then </b>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">b</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="c5">a</font>
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:13_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:13_2"><i><font color="Green" title="E24">A21</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c4">lambda</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span>;<br/>
<b>take </b><font color="Maroon" title="c5">u</font> = <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">u</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> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">u1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> holds <br/> ex <font color="Olive" title="b2">u2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">u2</font> &amp; <font color="Olive" title="b2">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> ) ) )</span><br/>

<a NAME="E6:13_2"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_4"/><i><font color="Green" title="E26">A26</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )</span><br/>

<b>assume </b><a NAME="E1:13_2_4_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_2_4_1"/><i><font color="Green" title="E27">A27</font></i>: 
( <font color="Maroon" title="c7">i</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="c3">ksi</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="#E3:13_2_4_1"><i><font color="Green" title="E27">A27</font></i></a></i>;<br/>
<a NAME="E5:13_2_4_1"/>
<font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:13_2_4_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="#E3:13_2_4_1"><i><font color="Green" title="E27">A27</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:13_2_4"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="card_1.html#T28" title="CARD_1:th.28">CARD_1:28</a>, <a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a></i>;<br/>
<a NAME="E3:13_2_4"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E4:13_2_4"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="#E1:13_2_4"><i><font color="Green" title="E26">A26</font></i></a>, <a class="ref" href="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a></i>;<br/>
<a NAME="E5:13_2_4"/><b>then </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T28" title="ZF_REFLE:th.28">ZF_REFLE:28</a></i>;<br/>
<b>hence </b><a NAME="E6:13_2_4"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">l</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font></span>)</span> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E8:13_2"/>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">l</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> &amp; <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:13_2"/>
( <font color="Maroon" title="c5">u</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> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> holds <br/> ex <font color="Olive" title="b2">u2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">u2</font> &amp; <font color="Olive" title="b2">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> )</span><br/>

<a NAME="E10:13_2"/><i><font color="Green" title="E26">A28</font></i>: 
<font color="Maroon" title="c6">l</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_5"/><i><font color="Green" title="E27">A29</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">l</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T5" title="ORDINAL2:th.5">ORDINAL2:5</a></i>;<br/>
<a NAME="E2:13_2_5"/>
<font color="Maroon" title="c6">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">l</font> or <font color="Maroon" title="c7">u1</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="c6">l</font> )</span><br/>

<b>assume </b><a NAME="E1:13_2_5_1"/>
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">u1</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="c6">l</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">u2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:13_2_5_1"/><i><font color="Green" title="E28">A30</font></i>: 
( <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font> &amp; <font color="Maroon" title="c7">u1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">u2</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:13_2_5_1"/><i><font color="Green" title="E29">A31</font></i>: 
( <font color="Maroon" title="c9">i</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="c3">ksi</font> &amp; <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <b>by </b><i><a class="txt" href="#E3:13_2_5_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="#E5:13_2_5_1"><i><font color="Green" title="E29">A31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">u3</font> = <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<a NAME="E8:13_2_5_1"/>
( <font color="Maroon" title="c11">u3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font> &amp; <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">u3</font> )
 
<b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="#E5:13_2"><i><font color="Green" title="E25">A25</font></i></a>, <a class="txt" href="#E5:13_2_5_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:13_2_5_1"/><b>then </b>
( <font color="Maroon" title="c11">u3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">u3</font> )
 
<b>by </b><i><a class="txt" href="#E3:13_2_5_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E10:13_2_5_1"/>
<font color="Maroon" title="c7">u1</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="c6">l</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:13_2_5"/><b>then </b>
<font color="Maroon" title="c6">l</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="c6">l</font>
 
<b>by </b><i><a class="txt" href="#E1:13_2_5"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:13_2_5"/>
<font color="Maroon" title="c6">l</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D6" title="ORDINAL1:def.6">ORDINAL1:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:13_2"/><i><font color="Green" title="E27">A32</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_6"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c7">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:13_2_6"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  <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="c6">l</font>
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">u1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  or <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="c6">l</font> )</span><br/>

<b>assume </b><a NAME="E1:13_2_6_1"/>
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c6">l</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_2_6_1"/><i><font color="Green" title="E28">A33</font></i>: 
( <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> &amp; <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:13_2_6_1"/><i><font color="Green" title="E29">A34</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:13_2_6_1"><i><font color="Green" title="E28">A33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<a NAME="E7:13_2_6_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font>
 
<b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:13_2_6_1"/><b>then </b>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">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="Maroon" title="c6">l</font>
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E9:13_2_6_1"/>
<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="c6">l</font>
 <b>by </b><i><a class="txt" href="#E3:13_2_6_1"><i><font color="Green" title="E28">A33</font></i></a>, <a class="txt" href="#E5:13_2_6_1"><i><font color="Green" title="E29">A34</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">u1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">l</font> or <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E2:13_2_6"/><i><font color="Green" title="E28">A35</font></i>: 
<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="c6">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> </span><br/>

<a NAME="E3:13_2_6"/>
 <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font>
 
<b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:13_2_6"/><b>then </b>
<font color="Maroon" title="c6">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E5:13_2_6"/><b>then </b>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E10:13_2"><i><font color="Green" title="E26">A28</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">u2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:13_2_6"/><i><font color="Green" title="E29">A36</font></i>: 
( <font color="Maroon" title="c9">u2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">l</font></span>)</span> &amp; <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u2</font> )
 <b>by </b><i><a class="txt" href="#E2:13_2_6"><i><font color="Green" title="E28">A35</font></i></a>, <a class="txt" href="zf_fund2.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<a NAME="E8:13_2_6"/><i><font color="Green" title="E30">A37</font></i>: 
( <font color="Maroon" title="c9">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">l</font> &amp; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u2</font> )
 
<b>by </b><i><a class="txt" href="#E7:13_2_6"><i><font color="Green" title="E29">A36</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E9:13_2_6"/><b>then </b><i><font color="Green" title="E31">A38</font></i>: 
( <font color="Maroon" title="c9">u2</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">L</font> &amp; <font color="Maroon" title="c9">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">l</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:13_2_6"/><b>then </b>
<font color="Maroon" title="c9">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#D5" title="ZF_REFLE:def.5">ZF_REFLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">u2</font> = <font color="Maroon" title="c9">u2</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">b</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E13:13_2_6"/><i><font color="Green" title="E32">A39</font></i>: 
( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ksi</font> &amp; <font color="Maroon" title="c10">u2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="txt" href="#E9:13_2_6"><i><font color="Green" title="E31">A38</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:13_2_6"/><i><font color="Green" title="E33">A40</font></i>: 
( <font color="Maroon" title="c12">i</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="c3">ksi</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 <b>by </b><i><a class="txt" href="#E13:13_2_6"><i><font color="Green" title="E32">A39</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="#E15:13_2_6"><i><font color="Green" title="E33">A40</font></i></a></i>;<br/>
<a NAME="E17:13_2_6"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 
<b>by </b><i><a class="txt" href="#E15:13_2_6"><i><font color="Green" title="E33">A40</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c11">b</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<a NAME="E19:13_2_6"/>
( <font color="Maroon" title="c10">u2</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c14">b</font> iff ( <font color="Maroon" title="c10">u2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c14">b</font> &amp; <font color="Maroon" title="c10">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E20:13_2_6"/><b>then </b>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c10">u2</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="c14">b</font>
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E13:13_2_6"><i><font color="Green" title="E32">A39</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<a NAME="E21:13_2_6"/><b>then </b>
( <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span> &amp; <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 
<b>by </b><i><a class="txt" href="#E8:13_2_6"><i><font color="Green" title="E30">A37</font></i></a>, <a class="txt" href="#E15:13_2_6"><i><font color="Green" title="E33">A40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:13_2_6"/>
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c7">u1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> implies  ex <font color="Olive" title="b1">u2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c7">u1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b1">u2</font> &amp; <font color="Olive" title="b1">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> ) )</span><br/>

<b>assume </b><a NAME="E12:13_2"/>
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">u2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c7">u1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b1">u2</font> &amp; <font color="Olive" title="b1">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c8">u2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:13_2"/><i><font color="Green" title="E28">A41</font></i>: 
( <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">u2</font> &amp; <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E11:13_2"><i><font color="Green" title="E27">A32</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">u2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">u1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c8">u2</font> &amp; <font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c9">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E16:13_2"/><i><font color="Green" title="E29">A42</font></i>: 
<font color="Maroon" title="c8">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:13_2"><i><font color="Green" title="E28">A41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<a NAME="E18:13_2"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c10">a</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>
 
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E19:13_2"/><b>then </b><i><font color="Green" title="E30">A43</font></i>: 
<font color="Maroon" title="c7">u1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">u2</font>
 
<b>by </b><i><a class="txt" href="#E14:13_2"><i><font color="Green" title="E28">A41</font></i></a>, <a class="txt" href="#E16:13_2"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E20:13_2"/>
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">u2</font>
 
<b>by </b><i><a class="txt" href="#E14:13_2"><i><font color="Green" title="E28">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:13_2"/>
<font color="Maroon" title="c7">u1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c8">u2</font>
 <b>by </b><i><a class="txt" href="#E19:13_2"><i><font color="Green" title="E30">A43</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c5">u</font></span><br/>

<b>thus </b><a NAME="E22:13_2"/>
<font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_7"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_7"/>
( <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 
<b>by </b><i><a class="txt" href="#E4:13_2"><i><font color="Green" title="E24">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:13_2_7"/>
<font color="Maroon" title="c8">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="txt" href="#E11:13_2"><i><font color="Green" title="E27">A32</font></i></a>, <a class="txt" href="#E16:13_2"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E13:13"/><b>then </b><i><font color="Green" title="E22">A44</font></i>: 
 <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#K9" title="ZF_MODEL:func.9">the_axiom_of_infinity</a> 
 
<b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="zfmodel1.html#T7" title="ZFMODEL1:th.7">ZFMODEL1:7</a></i>;<br/>
<a NAME="E14:13"/><i><font color="Green" title="E23">A45</font></i>: 
 <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> 
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="zf_fund2.html#T2" target="_self" title="ZF_FUND2:th.2">Th2</a></i>;<br/>
<a NAME="E15:13"/>
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Olive" title="b1">H</font> holds <br/> <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#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Olive" title="b1">H</font>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="zf_fund2.html#T3" target="_self" title="ZF_FUND2:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E16:13"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a>
 <b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="#E11:13"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="#E13:13"><i><font color="Green" title="E22">A44</font></i></a>, <a class="txt" href="#E14:13"><i><font color="Green" title="E23">A45</font></i></a>, <a class="ref" href="zf_model.html#D12" title="ZF_MODEL:def.12">ZF_MODEL:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
