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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msualg_2.html#V2" target="_self">all-with_const_op</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<a NAME="E1:30"/>
(  <a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="relat_1.html#V2">V5</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="relat_1.html#V2">V5</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E2:30"/><b>then </b>
(  <a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T23">PBOOLE:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:30_1"/>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/><a NAME="E4:30_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_2.html#E3:30"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:30_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>thus </b><a NAME="E7:30_1"/>
not <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="msualg_2.html#E4:30_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msualg_2.html#E6:30_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>


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