<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_1.html#V3">strict</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="E1:31"/><i><font color="Green">E25</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/>

<b>reconsider </b><font color="Maroon">c<sub>6</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#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as    <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="ref" href="mssublat.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E3:31"/>
 <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#K8">"\/"</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/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> 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> ;<br/>
<a NAME="E5:31"/>
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>4</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <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>;<br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</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#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E2:31_1"/>
<font color="Maroon">c<sub>7</sub></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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8">"\/"</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#K8">"\/"</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/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green">E27</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8">"\/"</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/>
<b>reconsider </b><font color="Maroon">c<sub>10</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="E5:31_1"/><i><font color="Green">E28</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>10</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="E6:31_1"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E7:31_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E6:31_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> 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> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = 0 as   <a href="struct_0.html#NM1">Element</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#E5:31_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:31_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:31_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>14</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/>

<b>then </b><i><font color="Green">E32</font></i>: <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#K2">\/</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>14</sub></font> = 
<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#K2">\/</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#E5:31_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</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#K2">\/</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="txt" href="mssublat.html#E1:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="mssublat.html#T28" target="_self">Th28</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#K2">\/</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="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</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#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E3:31_1"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<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#K8">"\/"</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#E5:31_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mssublat.html#E1:31_1_1_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#K8">"\/"</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/>
<a NAME="E4:31_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K7">GenUnivAlg</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D14">UNIALG_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1_1"/>
<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#K2">\/</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>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E2:31_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="mssublat.html#E3:31_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="unialg_2.html#D13">UNIALG_2:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E1:31_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:31_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <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> st <font color="Maroon">c<sub>8</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <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="E1:31_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_2_1"/>
( <font color="Maroon">c<sub>14</sub></font> is <a href="msualg_1.html#V5">non-empty</a> or  not <font color="Maroon">c<sub>14</sub></font> is <a href="msualg_1.html#V5">non-empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1"><b>suppose </b></a><a NAME="E1:31_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="msualg_1.html#V5">non-empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <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/><a NAME="E3:31_1_2_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="msualg_1.html#K13">1-Alg</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>
 <b>by </b><i><a class="ref" href="mssublat.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E4:31_1_2_1_1"/><b>then </b>
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="msualg_1.html#T15">MSUALG_1:15</a></i>;<br/><a NAME="E5:31_1_2_1_1"/><b>then </b><i><font color="Green">E33</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#K7">MSSign</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E6:31_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E1:31_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E7:31_1_2_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E8:31_1_2_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>15</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mssublat.html#E5:31_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mssublat.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E9:31_1_2_1_1"/>
 <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</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"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</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"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</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/><a NAME="E10:31_1_2_1_1"/><b>then </b>
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"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></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> <a href="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_1.html#D14">MSUALG_1:def 14</a></i>;<br/><a NAME="E11:31_1_2_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <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"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
 <b>by </b><i><a class="txt" href="mssublat.html#E8:31_1_2_1_1"><i><font color="Green">E35</font></i></a></i>;<br/><i><font color="Green">E37</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="mssublat.html#E1:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="mssublat.html#E1:31_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="mssublat.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E3:31_1_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="msualg_1.html#T15">MSUALG_1:15</a></i>;<br/><a NAME="E14:31_1_2_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E7:31_1_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="mssublat.html#E11:31_1_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="mssublat.html#E12:31_1_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><a NAME="E15:31_1_2_1_1"/><b>then </b>
 <a href="unialg_2.html#K7">GenUnivAlg</a> <font color="Maroon">c<sub>11</sub></font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D13">UNIALG_2:def 13</a></i>;<br/><a NAME="E16:31_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D14">UNIALG_2:def 14</a></i>;<br/><a NAME="E17:31_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E18:31_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>15</sub></font> #)
 <b>by </b><i><a class="txt" href="mssublat.html#E5:31_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mssublat.html#E5:31_1_2_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="mssublat.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E19:31_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="mssublat.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_2"><b>suppose </b></a><a NAME="E1:31_1_2_1_2"/>
not <font color="Maroon">c<sub>14</sub></font> is <a href="msualg_1.html#V5">non-empty</a>
 ;<br/><div class="add"><a NAME="E2:31_1_2_1_2"/><b>then </b>
not the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>14</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_1_2_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>15</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>
 <b>and </b><br/><a NAME="E5:31_1_2_1_2"/><i><font color="Green">E33</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><a NAME="E6:31_1_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E1:31_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E7:31_1_2_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E9:31_1_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><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">c<sub>2</sub></font></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="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <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">c<sub>2</sub></font> 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="txt" href="mssublat.html#E5:31_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E11:31_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 <b>by </b><i><a class="txt" href="mssublat.html#E1:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/><a NAME="E12:31_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#K2">\/</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="txt" href="mssublat.html#E1:31_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msualg_1.html#D14">MSUALG_1:def 14</a></i>;<br/><a NAME="E13:31_1_2_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>;<br/><a NAME="E14:31_1_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E15:31_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</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="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E16:31_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E13:31_1_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E17:31_1_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E5:31_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mssublat.html#E4:31_1_2_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="mssublat.html#E5:31_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mssublat.html#E7:31_1_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E13:31_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E11:31_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:31"/>
 <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#K8">"\/"</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#K13">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D19">MSUALG_2:def 19</a></i>;<br/>


</div>
