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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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><br/> for <font color="Olive" title="b2">G</font> being   <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">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> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font></span><br/><b>let </b><font color="Maroon" title="c2">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> holds <br/><font color="Maroon" title="c2">B</font> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font></span><br/><b>let </b><font color="Maroon" title="c3">G</font> be    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c2">B</font> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font> )</span><br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">B</font> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Olive" title="b1">n</font></span><br/><b>consider </b><font color="Maroon" title="c5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:53_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="aofa_000.html#T30" target="_self" title="AOFA_000:th.30">Th30</a></i>;<br/><b>take </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font></span><br/><a NAME="E3:53_1"/>
<font color="Maroon" title="c3">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="aofa_000.html#T22" target="_self" title="AOFA_000:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="#E2:53_1"><i><font color="Green" title="E34">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:53"/>
<font color="Maroon" title="c2">B</font> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="aofa_000.html#T30" target="_self" title="AOFA_000:th.30">Th30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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