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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>thus </b><a NAME="E1:2"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="osafree.html#M1" target="_self">OSGeneratorSet</a> of <font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="osalg_2.html#M2">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a>  <a href="osalg_2.html#K2">OSCl</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="osalg_2.html#K10">GenOSAlg</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <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">E2</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="osafree.html#M1" target="_self">OSGeneratorSet</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="osalg_2.html#M2">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:2_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a>  <a href="osalg_2.html#K2">OSCl</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
<a NAME="E4:2_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10">GenOSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:2_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="osafree.html#E2:2_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="osafree.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:2_1"/>
 <a href="osalg_2.html#K10">GenOSAlg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#T10">MSUALG_2:10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:2"/><i><font color="Green">E2</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="osalg_2.html#M2">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a>  <a href="osalg_2.html#K2">OSCl</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="osalg_2.html#K10">GenOSAlg</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="osalg_2.html#M2">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osafree.html#D1" target="_self">OSAFREE:def 1</a><br/>

<b>assume </b><a NAME="E3:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a>  <a href="osalg_2.html#K2">OSCl</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><a NAME="E4:2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10">GenOSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:2"><i><font color="Green">E2</font></i></a>, <a class="txt" href="osafree.html#E3:2"><i><font color="Green">E3</font></i></a></i>;<br/>


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