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<b>let </b><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</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">OU0</font> being  <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Olive" title="b1">OU0</font> holds  <span class="p1">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Olive" title="b1">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Olive" title="b2">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="Olive" title="b2">A</font></span><br/><b>let </b><font color="Maroon" title="c2">OU0</font> be   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</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> being   <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font> holds  <span class="p1">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Olive" title="b1">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="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <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><br/><b>let </b><font color="Maroon" title="c4">i</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="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</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="c4">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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/>







<b>assume </b><a NAME="E1:47"/>
<font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">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="E3:47"/><i><font color="Green" title="E28">A1</font></i>: 
<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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c5">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_2.html#D11" target="_self" title="OSALG_2:def.11">Def11</a></i>;<br/>
<a NAME="E4:47"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</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="c5">s</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Z</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"> ( <font color="Maroon" title="c6">Z</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="c5">s</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c6">Z</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="c5">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">B</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green" title="E29">A2</font></i>: 
( <font color="Maroon" title="c7">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> &amp; <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="ref" href="osalg_2.html#D10" target="_self" title="OSALG_2:def.10">Def10</a></i>;<br/>
<a NAME="E4:47_1"/>
(  <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="c7">B</font> &amp; <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">B</font> )
 
<b>by </b><i><a class="txt" href="osalg_2.html#E3:47_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="osalg_2.html#T24" target="_self" title="OSALG_2:th.24">Th24</a></i>;<br/>
<a NAME="E5:47_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></i>;<br/>
<b>hence </b><a NAME="E6:47_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E3:47_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</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>hence </b><a NAME="E5:47"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E3:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</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>
