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

<b>let </b><font color="Maroon">S</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">V</font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">X1</font> be    <a href="msaterm.html#M3">SetWithCompoundTerm</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>, <font color="Maroon">X2</font> be    <a href="msaterm.html#M3">SetWithCompoundTerm</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>







<b>thus </b><a NAME="E1:34"/>
( the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D1">CIRCCOMB:def 1</a>,<a class="ref" href="circcomb.html#D3">CIRCCOMB:def 3</a><br/>

<b>thus </b><a NAME="E2:34"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">x</font> as   <a href="msafree2.html#NM1">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">x</font> as   <a href="msafree2.html#NM1">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<b>thus </b>the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="circtrm1.html#K2" target="_self">the_sort_of</a> <font color="Maroon">v1</font>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="circtrm1.html#K2" target="_self">the_sort_of</a> <font color="Maroon">v2</font>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E3:34"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:34"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">x</font> as   <a href="circcomb.html#NM1">Gate</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">x</font> as   <a href="circcomb.html#NM1">Gate</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K1" target="_self">-CircuitStr</a> </span>)</span> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b>the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="circtrm1.html#K3" target="_self">the_action_of</a> <font color="Maroon">g1</font>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="circtrm1.html#K3" target="_self">the_action_of</a> <font color="Maroon">g2</font>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="circtrm1.html#K6" target="_self">-Circuit</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div>
