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<b>let </b><font color="Maroon">A1</font> be  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">A2</font> be  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/>the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/>the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>set </b><font color="Maroon">B</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font>;<br/>
<a NAME="E3:13_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><b>hence </b>the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E5:13_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/><b>assume </b><a NAME="E1:13_1_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E2:13_1_2_2"/><b>then </b><i><font color="Green">E45</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><i><font color="Green">E46</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, </i>
;<br/><b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">f1</font> = the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">f2</font> = the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:13_1_2_2_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:13_1_2_2_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:13_1_2_2_2"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:13_1_2_2"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:13_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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