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

<b>let </b><font color="Maroon" title="c1">F</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"> ( ( for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Olive" title="b1">X</font> is <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a> ) implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">F</font> is <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a> )</span><br/>

<b>assume </b><a NAME="E1:109"/><i><font color="Green" title="E25">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Olive" title="b1">X</font> is <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">F</font> is <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a></span><br/>

<b>let </b><font color="Maroon" title="c2">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="membered.html#D6" target="_self" title="MEMBERED:def.6">MEMBERED:def 6</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">F</font> implies <font color="Maroon" title="c2">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> )</span><br/>

<b>assume </b><a NAME="E2:109"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:109"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E5:109"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:109"/>
<font color="Maroon" title="c3">X</font> is <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>
 
<b>by </b><i><a class="txt" href="membered.html#E1:109"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="membered.html#E5:109"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:109"/>
<font color="Maroon" title="c2">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 <b>by </b><i><a class="txt" href="membered.html#E4:109"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="membered.html#D6" target="_self" title="MEMBERED:def.6">Def6</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>
