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

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font> be  <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>set </b><font color="Maroon" title="c4">S1</font> =  <a href="facirc_2.html#K1" title="FACIRC_2:func.1">SingleMSS</a> <font color="Maroon" title="c1">x</font>;<br/>
<b>set </b><font color="Maroon" title="c5">A1</font> =  <a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font>;<br/>
<b>assume </b><a NAME="E1:10"/>
<font color="Maroon" title="c1">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 <font color="Maroon" title="c2">S</font>
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green" title="E4">A1</font></i>: 
( <span class="p1">(<span class="default"><a href="facirc_2.html#K1" title="FACIRC_2:func.1">SingleMSS</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="facirc_2.html#T5" target="_self" title="FACIRC_2:th.5">Th5</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green" title="E5">A2</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" title="FACIRC_2:func.1">SingleMSS</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">--&gt;</a> <a href="circcomb.html#K10" title="CIRCCOMB:func.10">BOOLEAN</a>  &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" title="FACIRC_2:func.1">SingleMSS</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">--&gt;</a> <a href="circcomb.html#K10" title="CIRCCOMB:func.10">BOOLEAN</a>  )
 
<b>by </b><i><a class="ref" href="facirc_2.html#D1" target="_self" title="FACIRC_2:def.1">Def1</a>, <a class="ref" href="circcomb.html#T65" title="CIRCCOMB:th.65">CIRCCOMB:65</a></i>;<br/>
<a NAME="E4:10"/><b>then </b><i><font color="Green" title="E6">A3</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</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 <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#T4" title="CIRCCOMB:th.4">CIRCCOMB:4</a></i>;<br/>
<a NAME="E5:10"/>
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" title="FACIRC_2:func.1">SingleMSS</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#D5" title="MSUALG_1:def.5">MSUALG_1:def 5</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
 <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"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T21" title="FUNCT_4:th.21">FUNCT_4:21</a></i>;<br/>
<a NAME="E9:10"/><b>then </b><i><font color="Green" title="E7">A4</font></i>: 
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="facirc_2.html#E4:10"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="circcomb.html#D4" title="CIRCCOMB:def.4">CIRCCOMB:def 4</a></i>;<br/>
<a NAME="E10:10"/>
(  <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"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" title="FACIRC_2:func.1">SingleMSS</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <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 <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:10"/>
<span class="p1">(<span class="default"><a href="facirc_2.html#K2" title="FACIRC_2:func.2">SingleMSA</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c3">A</font> #)
 <b>by </b><i><a class="txt" href="facirc_2.html#E2:10"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="facirc_2.html#E3:10"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="facirc_2.html#E9:10"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="funct_4.html#T20" title="FUNCT_4:th.20">FUNCT_4:20</a></i>;<br/>


</div>
