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

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</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">A1</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A2</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S2</font>;<br/>





<b>set </b><font color="Maroon">A</font> = <font color="Maroon">A1</font> <a href="circcomb.html#K4" target="_self">+*</a> <font color="Maroon">A2</font>;<br/>
<b>set </b><font color="Maroon">S</font> = <font color="Maroon">S1</font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">S2</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:111"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">A1</font> is <a href="circcomb.html#V4" target="_self">gate`2=den</a> &amp; <font color="Maroon">A2</font> is <a href="circcomb.html#V4" target="_self">gate`2=den</a> )
 <b>and </b><br/><a NAME="E2:111"/><i><font color="Green">E33</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E3:111"/><i><font color="Green">E38</font></i>: 
the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4" target="_self">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="circcomb.html#K2" target="_self">+*</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E4:111"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:111"/><i><font color="Green">E43</font></i>: 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="circcomb.html#T5" target="_self">Th5</a></i>;<br/>
<b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D10" target="_self">CIRCCOMB:def 10</a><br/>

<b>assume </b><a NAME="E6:111"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">S2</font></span>)</span>
 ;<br/>

<a NAME="E7:111"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:111"/><b>then </b>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> or <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:111"/><b>then </b>
( ( the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4" target="_self">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) or ( the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4" target="_self">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) )
 
<b>by </b><i>, <a class="ref" href="circcomb.html#T9" target="_self">Th9</a>, , , , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="funct_4.html#T16">FUNCT_4:16</a></i>;<br/>
<b>hence </b><a NAME="E10:111"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <span class="p3">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4" target="_self">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div>
