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<b>let </b><font color="Maroon">A</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">a1</font> be  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">A</font>, <font color="Maroon">b1</font> be  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">a2</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A</font>, <font color="Maroon">b2</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:29"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">b1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E33</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<font color="Maroon">ff1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16">MSUALG_1:16</a></i>;<br/>
<a NAME="E4:29"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8">MSSorts</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">a1</font></span>)</span> #)
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E34</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D14">MSUALG_1:def 14</a></i>;<br/>
<a NAME="E6:29"/>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8">MSSorts</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">b1</font></span>)</span> #)
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/>
<a NAME="E7:29"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D14">MSUALG_1:def 14</a></i>;<br/>
<a NAME="E8:29"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><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="E1:29_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:29_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<span class="p1">(<span class="default">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">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E4:29_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">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">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:29"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">a2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>;<br/>


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