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<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Olive" title="b2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> holds <br/>( <font color="Olive" title="b2">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">B1</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Olive" title="b2">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">B1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )</span><br/><b>let </b><font color="Maroon" title="c2">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Olive" title="b1">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> holds <br/>( <font color="Olive" title="b1">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Olive" title="b1">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )</span><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</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">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c3">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> implies ( <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> ) )</span><br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <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> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> or  not <font color="Maroon" title="c3">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> or ( <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> ) )</span><br/>

<b>assume </b><a NAME="E2:20"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c3">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c3">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</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">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )</span><br/>

<a NAME="E3:20"/><i><font color="Green" title="E20">A3</font></i>: 
for <font color="Olive" title="b1">B2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/> ex <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Olive" title="b1">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">B2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies  ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> ) )</span><br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</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">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1"/>
( <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> or <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B1</font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</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">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c5">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:20_1_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c5">D</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="E4:20_1_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E2:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="card_lar.html#E1:20_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font> )</span><br/><a NAME="E5:20_1_1_1"/>
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:20_1_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="card_lar.html#E4:20_1_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:20_1_1_1"/>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E3:20_1_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font></span><br/><b>thus </b><a NAME="E7:20_1_1_1"/>
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E4:20_1_1_1"><i><font color="Green" title="E24">A7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B1</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">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c5">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:20_1_1_2"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c5">D</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="E4:20_1_1_2"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c2">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="card_lar.html#E2:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font> )</span><br/><a NAME="E5:20_1_1_2"/>
not <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B1</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E4:20_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E6:20_1_1_2"/>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E3:20_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font></span><br/><b>thus </b><a NAME="E7:20_1_1_2"/>
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:20_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="card_lar.html#E4:20_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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>

</div><b>end;</b></div>
<b>thus </b><a NAME="E4:20"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</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">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">B</font> be  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">CARD_LAR:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> )</span><br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> )</span><br/>

<b>assume </b><a NAME="E2:20_2"/><i><font color="Green" title="E22">A12</font></i>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span><br/>

<a NAME="E3:20_2"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E2:20_2_1"/><b>then </b><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:20_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="ordinal2.html#T30" title="ORDINAL2:th.30">ORDINAL2:30</a></i>;<br/>
<a NAME="E3:20_2_1"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T30" title="ORDINAL2:th.30">ORDINAL2:30</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:20_2_1"/><b>then </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></i>;<br/>
<b>hence </b><a NAME="E5:20_2_1"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E2:20_2_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0: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="E4:20_2"/><b>then </b><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="card_lar.html#E1:20_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">Def5</a></i>;<br/>
<b>assume </b><a NAME="E5:20_2"/>
not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:20_2"/><b>then </b>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:20_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:20_2"/><b>then </b><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:20_2"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E9:20_2"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:20_2"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E10:20_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:20_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E2:20_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</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>thus </b><a NAME="E5:20"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 <b>by </b><i><a class="txt" href="card_lar.html#E3:20"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</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>
