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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="circcomb.html#V6">Boolean</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="facirc_2.html#K1" target="_self">SingleMSS</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:10"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green">E4</font></i>: 
( <span class="p1">(<span class="default"><a href="facirc_2.html#K1" target="_self">SingleMSS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="facirc_2.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green">E5</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" target="_self">SingleMSS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="circcomb.html#K5">--&gt;</a> <a href="circcomb.html#K10">BOOLEAN</a>  &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" target="_self">SingleMSS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="circcomb.html#K10">BOOLEAN</a>  )
 
<b>by </b><i><a class="ref" href="facirc_2.html#D1" target="_self">Def1</a>, <a class="ref" href="circcomb.html#T65">CIRCCOMB:65</a></i>;<br/>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E6</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="circcomb.html#T4">CIRCCOMB:4</a></i>;<br/>
<a NAME="E5:10"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" target="_self">SingleMSS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#D5">MSUALG_1:def 5</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="circcomb.html#K1">+*</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T21">FUNCT_4:21</a></i>;<br/>
<a NAME="E9:10"/><b>then </b><i><font color="Green">E7</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="circcomb.html#K4">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="circcomb.html#K1">+*</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="circcomb.html#K4">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="facirc_2.html#E4:10"><i><font color="Green">E6</font></i></a>, <a class="ref" href="circcomb.html#D4">CIRCCOMB:def 4</a></i>;<br/>
<a NAME="E10:10"/>
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K2" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_2.html#K1" target="_self">SingleMSS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">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" target="_self">SingleMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="circcomb.html#K4">+*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
 <b>by </b><i><a class="txt" href="facirc_2.html#E2:10"><i><font color="Green">E4</font></i></a>, <a class="txt" href="facirc_2.html#E3:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="facirc_2.html#E9:10"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_4.html#T20">FUNCT_4:20</a></i>;<br/>


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