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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:42"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:42"/><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="tarski.html#R2" title="TARSKI: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" onmouseover="rs('osalg_2/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E3:42"/>
 <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#T16" target="_self" onmouseover="rs('osalg_2/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:42"/>
<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="#E2:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="msualg_2.html#D13" onmouseover="rs('msualg_2/D13')" onmouseout="rh()">MSUALG_2:def 13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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