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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be    <a href="msaterm.html#M3" title="MSATERM:mode.3">SetWithCompoundTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c5">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/>







<b>thus </b><a NAME="E1:34"/>
( the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="circtrm1.html#T7" target="_self" title="CIRCTRM1:th.7">Th7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D1" title="CIRCCOMB:def.1">CIRCCOMB:def 1</a>,<a class="ref" href="circcomb.html#D3" title="CIRCCOMB:def.3">CIRCCOMB:def 3</a><br/>

<b>thus </b><a NAME="E2:34"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c6">x</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <b>by </b><i><a class="txt" href="circtrm1.html#E2:34_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">v2</font> = <font color="Maroon" title="c6">x</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <b>by </b><i><a class="txt" href="circtrm1.html#E2:34_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">t</font> = <font color="Maroon" title="c6">x</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="circtrm1.html#E3:34_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="circtrm1.html#T4" target="_self" title="CIRCTRM1:th.4">Th4</a></i>;<br/>
<b>thus </b>the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
 <a href="circtrm1.html#K2" title="CIRCTRM1:func.2">the_sort_of</a> <font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="circtrm1.html#D4" target="_self" title="CIRCTRM1:def.4">Def4</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c9">t</font></span>)</span>
<b>by </b><i><a class="ref" href="circtrm1.html#D2" target="_self" title="CIRCTRM1:def.2">Def2</a></i>
<br/>.= 
 <a href="circtrm1.html#K2" title="CIRCTRM1:func.2">the_sort_of</a> <font color="Maroon" title="c8">v2</font>,<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="circtrm1.html#D2" target="_self" title="CIRCTRM1:def.2">Def2</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="circtrm1.html#D4" target="_self" title="CIRCTRM1:def.4">Def4</a></i>
;<br/>


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

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E3:34"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:34"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c6">x</font> as   <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g2</font> = <font color="Maroon" title="c6">x</font> as   <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <b>by </b><i><a class="txt" href="circtrm1.html#E4:34"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b>the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
 <a href="circtrm1.html#K3" title="CIRCTRM1:func.3">the_action_of</a> <font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="circtrm1.html#D5" target="_self" title="CIRCTRM1:def.5">Def5</a></i>
<br/>.= 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="circtrm1.html#D3" target="_self" title="CIRCTRM1:def.3">Def3</a></i>
<br/>.= 
 <a href="circtrm1.html#K3" title="CIRCTRM1:func.3">the_action_of</a> <font color="Maroon" title="c8">g2</font>,<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="circtrm1.html#D3" target="_self" title="CIRCTRM1:def.3">Def3</a></i>
<br/>.= 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="circtrm1.html#D5" target="_self" title="CIRCTRM1:def.5">Def5</a></i>
;<br/>


</div>
