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<b>let </b><font color="Maroon" title="c1">I</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">M</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font>, <font color="Olive" title="b4">C</font> being  <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b2">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Olive" title="b4">C</font> holds <br/><font color="Olive" title="b2">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Olive" title="b4">C</font></span><br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <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">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being  <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b2">B</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Olive" title="b3">C</font> holds <br/><font color="Olive" title="b1">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Olive" title="b3">C</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</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">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c4">B</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Maroon" title="c5">C</font> implies <font color="Maroon" title="c3">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Maroon" title="c5">C</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Maroon" title="c5">C</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">A</font> <a href="closure3.html#R1" title="CLOSURE3:pred.1">is_finer_than</a> <font color="Maroon" title="c5">C</font></span><br/>

<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="closure3.html#D1" target="_self" title="CLOSURE3:def.1">CLOSURE3:def 1</a> <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="c3">A</font> implies  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> ) )</span><br/>

<b>assume </b><a NAME="E3:5"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:5"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E6:5"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="closure3.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="closure3.html#D1" target="_self" title="CLOSURE3:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:5"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 <b>and </b><br/><a NAME="E9:5"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="closure3.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="closure3.html#E5:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="closure3.html#D1" target="_self" title="CLOSURE3:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">c</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font> )</span><br/>

<b>thus </b><a NAME="E10:5"/>
( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="closure3.html#E6:5"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="closure3.html#E8:5"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="closure3.html#E9:5"><i><font color="Green" title="E10">A6</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/>


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