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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E27</font></i>: 
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="unialg_2.html#K6" target="_self">Constants</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green">E28</font></i>: 
<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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D13" target="_self">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> 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="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</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> ;<br/>
<a NAME="E5:41"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E7:41"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E2:41"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E8:41"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="txt" href="unialg_2.html#E6:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:41"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font> is  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="ref" href="unialg_2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E10:41"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="ref" href="unialg_2.html#D13" target="_self">Def13</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E11:41"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:41"/><b>then </b><i><font color="Green">E33</font></i>: 
<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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E2:41"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="unialg_2.html#K6" target="_self">Constants</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i><a class="txt" href="unialg_2.html#E1:41"><i><font color="Green">E27</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>case </b></a><a NAME="E1:41_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="unialg_2.html#E12:41"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>case </b></a><a NAME="E1:41_1_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="unialg_2.html#K6" target="_self">Constants</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:41_1_1_2"/>
(  <a href="unialg_2.html#K6" target="_self">Constants</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="unialg_2.html#K6" target="_self">Constants</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="unialg_2.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:41_1_1_2"/><b>then </b>
 <a href="unialg_2.html#K6" target="_self">Constants</a> <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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><b>hence </b><a NAME="E4:41_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:41_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E14:41"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E15:41"/><b>then </b><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</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="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E16:41"/><b>then </b>
 <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="txt" href="unialg_2.html#E12:41"><i><font color="Green">E33</font></i></a>, <a class="ref" href="unialg_2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E17:41"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E18:41"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E19:41"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E15:41"><i><font color="Green">E34</font></i></a>, <a class="txt" href="unialg_2.html#E17:41"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E20:41"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E21:41"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E11:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E22:41"/><b>then </b>
 <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>7</sub></font> is  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of  <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E23:41"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E5:41"><i><font color="Green">E29</font></i></a>, <a class="txt" href="unialg_2.html#E9:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="unialg_2.html#T13" target="_self">Th13</a></i>;<br/>


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