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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_2.html#V2">with_const_op</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:33"/><i><font color="Green">E27</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:33"/>
 <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/>
<b>then </b><i><font color="Green">E28</font></i>: the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D14">MSUALG_1:def 14</a></i>
<br/>.= 
0 <a href="cqc_lang.html#K3">.--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></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">c<sub>1</sub></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="E6:33"/><i><font color="Green">E29</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></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="E7:33"/><b>then </b>
 <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="msualg_1.html#K10">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E4:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:33"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:33_2_1"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#R2">meets</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="ref" href="unialg_2.html#T20">UNIALG_2:20</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="mssublat.html#E4:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mssublat.html#E6:33"><i><font color="Green">E29</font></i></a>, <a class="txt" href="mssublat.html#E1:33_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</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="mssublat.html#E2:33_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="unialg_2.html#D10">UNIALG_2:def 10</a></i>
<br/>.= 
<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">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<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">c<sub>4</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="mssublat.html#E1:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="mssublat.html#T29" target="_self">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E6:33"><i><font color="Green">E29</font></i></a>, <a class="txt" href="mssublat.html#E1:33_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as  <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">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E2:33"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mssublat.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E11:33"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>8</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E12:33"/>
 <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_2.html#K11">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E9:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msualg_2.html#D17">MSUALG_2:def 17</a></i>;<br/>


</div>
