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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">U0</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <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">U1</font> being   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Olive" title="b1">U0</font><br/>  for <font color="Olive" title="b3">B</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Olive" title="b1">U0</font>  st <font color="Olive" title="b3">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">U0</font> holds <br/><span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Olive" title="b2">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b3">B</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="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">U1</font> being   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font><br/>  for <font color="Olive" title="b2">B</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Olive" title="b2">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font> holds <br/><span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Olive" title="b1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">U1</font> be   <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">U0</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">B</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Olive" title="b1">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font> holds <br/><span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c3">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b1">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</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="c4">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font> implies <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c3">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">B</font> )</span><br/>







<a NAME="E1:65"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</font> is   <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#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:65"/>
<font color="Maroon" title="c4">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</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"> <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c3">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">B</font></span><br/>

<a NAME="E3:65"/><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> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="pboole.html#D18" onmouseover="rs('pboole/D18')" onmouseout="rh()">PBOOLE:def 18</a></span>;<br/>
<a NAME="E4:65"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</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#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T22" onmouseover="rs('pboole/T22')" onmouseout="rh()">PBOOLE:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:65"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c3">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T37" target="_self" onmouseover="rs('osalg_2/T37')" onmouseout="rh()">Th37</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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