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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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><br/> for <font color="Olive" title="b3">s</font> being  <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds   <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">s</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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><br/> for <font color="Olive" title="b2">s</font> being  <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds   <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">s</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds   <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</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">  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c5">x</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="c4">s</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:47"/>
<font color="Maroon" title="c5">x</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="c4">s</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s</font></span><br/>

<a NAME="E2:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 ex <font color="Olive" title="b1">B</font> being   <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font> st <br/>( <font color="Olive" title="b1">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="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D10" target="_self" title="OSALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E3:47"/>
 <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_2.html#K5" title="MSUALG_2:func.5">SubSort</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T21" target="_self" title="OSALG_2:th.21">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:47"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="msualg_2.html#D14" title="MSUALG_2:def.14">MSUALG_2:def 14</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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