<?xml version="1.0"?>
<div class="add">

<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  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>3</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</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="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:28"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:28"/><i><font color="Green">E18</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="circcomb.html#E2"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:28"/><i><font color="Green">E19</font></i>: 
( the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_4.html#K1">+*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_4.html#K1">+*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="circcomb.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b> <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
the <a href="msualg_1.html#U2">Arity</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>
<b>by </b><i><a class="txt" href="circcomb.html#E1:28"><i><font color="Green">E17</font></i></a>, <a class="txt" href="circcomb.html#E2:28"><i><font color="Green">E18</font></i></a>, <a class="txt" href="circcomb.html#E3:28"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>

<b>thus </b> <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</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>
<b>by </b><i><a class="txt" href="circcomb.html#E1:28"><i><font color="Green">E17</font></i></a>, <a class="txt" href="circcomb.html#E2:28"><i><font color="Green">E18</font></i></a>, <a class="txt" href="circcomb.html#E3:28"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>


</div>
