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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>  <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/>



<a NAME="E1:103"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:103"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="circcomb.html#V6" target="_self">Boolean</a> implies the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="circcomb.html#K10" target="_self">BOOLEAN</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:103_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> holds the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K10" target="_self">BOOLEAN</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D15" target="_self">CIRCCOMB:def 15</a><br/>

<a NAME="E2:103_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <br/>the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K10" target="_self">BOOLEAN</a> 
 
;<br/>
<b>hence </b><a NAME="E3:103_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="circcomb.html#K10" target="_self">BOOLEAN</a> 
 <b>by </b><i><a class="txt" href="circcomb.html#E1:103"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>


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

<b>assume </b><a NAME="E3:103"/><i><font color="Green">E60</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="circcomb.html#K10" target="_self">BOOLEAN</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D15" target="_self">CIRCCOMB:def 15</a><br/>

<b>thus </b><a NAME="E4:103"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K10" target="_self">BOOLEAN</a> 
 <b>by </b><i><a class="txt" href="circcomb.html#E3:103"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


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