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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:10"/>
( <font color="Maroon" title="c2">G</font> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> &amp; <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="freealg.html#D5" target="_self" title="FREEALG:def.5">FREEALG:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c3">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c2">G</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E3:10_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="unialg_2.html#D13" title="UNIALG_2:def.13">UNIALG_2:def 13</a></i>;<br/>
<a NAME="E4:10_1"/>
<font color="Maroon" title="c3">C</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="freealg.html#E1:10_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="freealg.html#E3:10_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:10"/><i><font color="Green" title="E8">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="freealg.html#D5" target="_self" title="FREEALG:def.5">FREEALG:def 5</a><br/>

<b>assume </b><a NAME="E4:10"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c3">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> &amp; <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>

<b>thus </b><a NAME="E5:10"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>reconsider </b><font color="Maroon" title="c4">C</font> = <font color="Maroon" title="c3">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="freealg.html#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="freealg.html#E4:10"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="freealg.html#E7"><i><font color="Green" title="E5">Lm2</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E7:10"/><i><font color="Green" title="E10">A6</font></i>: 
 <a href="unialg_2.html#K4" title="UNIALG_2:func.4">UniAlgSetClosed</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">C</font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="freealg.html#E4:10"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="unialg_2.html#D9" title="UNIALG_2:def.9">UNIALG_2:def 9</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
 <a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c2">G</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_2.html#K4" title="UNIALG_2:func.4">UniAlgSetClosed</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="freealg.html#E4:10"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="unialg_2.html#D13" title="UNIALG_2:def.13">UNIALG_2:def 13</a></i>;<br/>
<a NAME="E9:10"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E3:10"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="freealg.html#E7:10"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:10"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 ;<br/>


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