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<span class="kw">set </span><font color="Maroon" title="c4">a</font> =  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">u1</font> = <font color="Maroon" title="c2">OU0</font> <a href="msualg_2.html#K10" title="MSUALG_2:func.10">|</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span> as   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">u1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c5">u1</font> &amp; ( for <font color="Olive" title="b1">OU1</font> being  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font>  st <font color="Maroon" title="c3">A</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Olive" title="b1">OU1</font> holds <br/><font color="Maroon" title="c5">u1</font> is   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Olive" title="b1">OU1</font> ) )</span><br/>

<a NAME="E2:61_1_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">OU0</font>,<span class="p2">(<span class="default"><a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D16" title="MSUALG_2:def.16">MSUALG_2:def 16</a></span>;<br/>
<a NAME="E3:61_1_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">A</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></span>;<br/>
<a NAME="E4:61_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T34" target="_self" title="OSALG_2:th.34">Th34</a></span>;<br/>
<a NAME="E5:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:61_1_1"/>
<font color="Maroon" title="c3">A</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c5">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">OU1</font> being  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font>  st <font color="Maroon" title="c3">A</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Olive" title="b1">OU1</font> holds <br/><font color="Maroon" title="c5">u1</font> is   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Olive" title="b1">OU1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">U2</font> be   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c6">U2</font> implies <font color="Maroon" title="c5">u1</font> is   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c6">U2</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">B1</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U2</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">OU0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></span>;<br/>
<a NAME="E8:61_1_1"/>
<font color="Maroon" title="c7">B1</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">B1</font> as    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:61_1_1"/>
<font color="Maroon" title="c3">A</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c6">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">u1</font> is   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c6">U2</font></span><br/>

<a NAME="E11:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A3</font></span>: 
( <font color="Maroon" title="c8">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c8">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E12:61_1_1"/>
 <a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c6">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T18" target="_self" title="OSALG_2:th.18">Th18</a></span>;<br/>
<a NAME="E13:61_1_1"/><span class="kw">then </span>
 <a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E14:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="osalg_2.html#T24" target="_self" title="OSALG_2:th.24">Th24</a></span>;<br/>
<a NAME="E15:61_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">u1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> or the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c9">i</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> ;<br/>
<a NAME="E3:61_1_1_1"/>
( the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c10">s</font></span>)</span> &amp; <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c10">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E14:61_1_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="osalg_2.html#D10" target="_self" title="OSALG_2:def.10">Def10</a>, <a class="ref" href="osalg_2.html#D11" target="_self" title="OSALG_2:def.11">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:61_1_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:61_1_1"/>
<font color="Maroon" title="c5">u1</font> is   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c6">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T9" title="MSUALG_2:th.9">MSUALG_2:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
