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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/>(<font color="Olive" title="b1">a</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> (<font color="Olive" title="b2">b</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> implies (<font color="Maroon" title="c2">a</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> (<font color="Maroon" title="c3">b</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (<font color="Maroon" title="c2">a</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> (<font color="Maroon" title="c3">b</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:18"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
(<font color="Maroon" title="c2">a</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> (<font color="Maroon" title="c3">b</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> )</span><br/><a NAME="E1:18_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> iff <span class="p1">(<span class="default">(<font color="Maroon" title="c3">b</font>,1) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="uproots.html#T6" title="UPROOTS:th.6">UPROOTS:6</a>, <a class="ref" href="uproots.html#T7" title="UPROOTS:th.7">UPROOTS:7</a></span>;<br/><span class="kw">hence </span><a NAME="E2:18_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T6" title="UPROOTS:th.6">UPROOTS:6</a>, <a class="ref" href="uproots.html#T7" title="UPROOTS:th.7">UPROOTS:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:18"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
