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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="msuhom_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="msuhom_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:7"/><i><font color="Green">E7</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <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> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:7"><i><font color="Green">E5</font></i></a>, <a class="ref" href="msuhom_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green">E8</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="msuhom_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:7"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E3:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msuhom_1.html#D1" target="_self">MSUHOM_1:def 1</a><br/>

<b>thus </b><a NAME="E6:7"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E3:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="msuhom_1.html#E4:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>thus </b>the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> = 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msuhom_1.html#E3:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="msuhom_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:7"><i><font color="Green">E5</font></i></a>, <a class="ref" href="msuhom_1.html#D1" target="_self">Def1</a></i>
;<br/>

<b>thus </b>the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> = 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msuhom_1.html#E3:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="msuhom_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:7"><i><font color="Green">E5</font></i></a>, <a class="ref" href="msuhom_1.html#D1" target="_self">Def1</a></i>
;<br/>


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