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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <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">U0</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">U1</font>, <font color="Olive" title="b3">U2</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">U0</font> holds  <font color="Olive" title="b2">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Olive" title="b3">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">U1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">U0</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <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">U1</font>, <font color="Olive" title="b2">U2</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> holds  <font color="Olive" title="b1">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Olive" title="b2">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">U1</font>, <font color="Maroon" title="c4">U2</font> be   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</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">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U1</font></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">u1</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</font>, <font color="Maroon" title="c6">u2</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">U2</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E2:60"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D17" target="_self" title="MSUALG_2:def.17">Def17</a></span>;<br/>
<a NAME="E3:60"/>
( <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="c2">U0</font> &amp; <font color="Maroon" title="c6">u2</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="c2">U0</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="E4:60"/><span class="kw">then </span>
<font color="Maroon" title="c5">u1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c6">u2</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="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c5">u1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <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">U0</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="E6:60"/>
<font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D19" target="_self" title="MSUALG_2:def.19">Def19</a></span>;<br/>
<a NAME="E7:60"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D18" target="_self" title="MSUALG_2:def.18">Def18</a></span>;<br/>
<a NAME="E8:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c7">A</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span>
 
<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="E9:60"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E10:60"/><span class="kw">then </span>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E11:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A3</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="pboole.html#T19" title="PBOOLE:th.19">PBOOLE:19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">u112</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E13:60"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c8">u112</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D17" target="_self" title="MSUALG_2:def.17">Def17</a></span>;<br/>
<a NAME="E14:60"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> <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="c3">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="pboole.html#T17" title="PBOOLE:th.17">PBOOLE:17</a></span>;<br/>
<a NAME="E15:60"/><span class="kw">then </span>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:60"/>
<font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c4">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</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/>


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