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

<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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <font color="Maroon" title="c1">A</font> iff for <font color="Olive" title="b2">o</font> being   <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font>  holds not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">o</font> )</span><br/>

<b>set </b><font color="Maroon" title="c2">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font></span>)</span> where <font color="Olive" title="b1">o</font> is    <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> : verum </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <font color="Maroon" title="c1">A</font> iff for <font color="Olive" title="b1">o</font> being   <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font>  holds not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font> )</span><br/>

<div><b>hereby </b> <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">o</font> being   <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font>  holds not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font> ) implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <font color="Maroon" title="c1">A</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <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">o</font> being   <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font>  holds not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font></span><br/><a NAME="E2:48_1"/><b>then </b><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font></span>)</span> where <font color="Olive" title="b1">o</font> is    <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>given </b><font color="Maroon" title="c4">o</font> being    <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font><b> such that </b><a NAME="E4:48_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E5:48_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font></span>)</span> where <font color="Olive" title="b1">o</font> is    <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> : verum </span>}</span> 
 ;<br/><b>hence </b><a NAME="E6:48_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:48_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="aofa_000.html#E4:48_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>assume </b><a NAME="E2:48"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">o</font> being   <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font>  holds not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <font color="Maroon" title="c1">A</font></span><br/>

<b>assume </b><a NAME="E3:48"/>
<font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <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"> contradiction</span><br/>

<a NAME="E4:48"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font></span>)</span> where <font color="Olive" title="b1">o</font> is    <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:48"/><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font></span>)</span> where <font color="Olive" title="b1">o</font> is    <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:48"/>
 ex <font color="Olive" title="b1">o</font> being   <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E6:48"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:48"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:48"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="aofa_000.html#E6:48"><i><font color="Green" title="E31">A4</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>
