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<b>let </b><font color="Maroon" title="c1">S</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">U0</font> being <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <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">S</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Olive" title="b1">U0</font> holds <br/> ( <font color="Olive" title="b2">A</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of <font color="Olive" title="b1">U0</font> iff for <font color="Olive" title="b3">O</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">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Olive" title="b2">A</font> holds <br/> <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b3">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U0</font> )</span><br/><b>let </b><font color="Maroon" title="c2">U0</font> be  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <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">S</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="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> holds <br/> ( <font color="Olive" title="b1">A</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of <font color="Maroon" title="c2">U0</font> iff for <font color="Olive" title="b2">O</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">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Olive" title="b1">A</font> holds <br/> <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font> )</span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of <font color="Maroon" title="c2">U0</font> iff for <font color="Olive" title="b1">O</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">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font> )</span><br/>





<b>thus </b><a NAME="E1:2"/>
( <font color="Maroon" title="c3">A</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of <font color="Maroon" title="c2">U0</font> implies for <font color="Olive" title="b1">O</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">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</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">O</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">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font> ) implies <font color="Maroon" title="c3">A</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of <font color="Maroon" title="c2">U0</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c3">A</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of <font color="Maroon" title="c2">U0</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">O</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">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font></span><br/>

<b>let </b><font color="Maroon" title="c4">O</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font> implies  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font> )</span><br/>

<b>assume </b><a NAME="E2:2_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c4">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">U1</font> = <font color="Maroon" title="c2">U0</font> as    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#T6" title="MSUALG_2:th.6">MSUALG_2:6</a></i>;<br/>
<a NAME="E4:2_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">O</font></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="c5">U1</font>
 
<b>by </b><i><a class="txt" href="#E1:2_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E2:2_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="osafree.html#D1" target="_self" title="OSAFREE:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:2_1"/>
 <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#T10" title="MSUALG_2:th.10">MSUALG_2:10</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>assume </b><a NAME="E2:2"/><i><font color="Green" title="E2">A3</font></i>: 
for <font color="Olive" title="b1">O</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">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of <font color="Maroon" title="c2">U0</font></span><br/>

<b>let </b><font color="Maroon" title="c4">O</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osafree.html#D1" target="_self" title="OSAFREE:def.1">OSAFREE:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font> implies the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">O</font></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="c2">U0</font> )</span><br/>

<b>assume </b><a NAME="E3:2"/><i><font color="Green" title="E3">A4</font></i>: 
<font color="Maroon" title="c4">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">O</font></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="c2">U0</font></span><br/>

<b>thus </b><a NAME="E4:2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c4">O</font></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="c2">U0</font>
 <b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E2">A3</font></i></a>, <a class="txt" href="#E3:2"><i><font color="Green" title="E3">A4</font></i></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>
