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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_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <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> be   <a href="msualg_2.html#NM1">MSSubset</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="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>







<b>assume </b><a NAME="E1:75"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K5" target="_self">SubSort</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:75"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a> &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> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="msualg_2.html#E1:75"><i><font color="Green">E53</font></i></a>, <a class="ref" href="msualg_2.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E3:75"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E1:75"><i><font color="Green">E53</font></i></a>, <a class="ref" href="msualg_2.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:75"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K6" target="_self">SubSort</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msualg_2.html#E2:75"><i><font color="Green">E54</font></i></a>, <a class="ref" href="msualg_2.html#D12" target="_self">Def12</a></i>;<br/>


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