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<a NAME="E1:25_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
 <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">MA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_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="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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">MA</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font> )</span><br/>

<b>assume </b><a NAME="E1:25_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">MA</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font></span><br/>

<a NAME="E2:25_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D12" title="MSUALG_2:def.12">MSUALG_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font>
 <b>by </b><i><a class="ref" href="closure2.html#D1" title="CLOSURE2:def.1">CLOSURE2:def 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>
<b>reconsider </b><font color="Maroon" title="c3">SS</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="struct_0.html#G1" title="STRUCT_0:aggr.1">1-sorted</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> #) ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">SF</font> =  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">MA</font> as   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c3">SS</font> <b>by </b><i><a class="txt" href="closure3.html#E1:25_1_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>
<b>take </b>
 <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c3">SS</font>,<font color="Maroon" title="c4">SF</font> #)
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c3">SS</font>,<font color="Maroon" title="c4">SF</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font> &amp; the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c3">SS</font>,<font color="Maroon" title="c4">SF</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">MA</font> )</span><br/>

<b>thus </b><a NAME="E4:25_1_1"/>
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c3">SS</font>,<font color="Maroon" title="c4">SF</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">MA</font> &amp; the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c3">SS</font>,<font color="Maroon" title="c4">SF</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">MA</font> )
 ; <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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