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<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">u</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>, <font color="Maroon">PB</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>







<b>assume </b><a NAME="E1:21"/>
<font color="Maroon">u</font> <a href="bvfunc_2.html#R3">is_independent_of</a> <font color="Maroon">PB</font>,<font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">u</font> <a href="bvfunc_1.html#R2">is_dependent_of</a>  <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PB</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D8">BVFUNC_2:def 8</a></i>;<br/>
<a NAME="E3:21"/><i><font color="Green">E20</font></i>: 
 <a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green">E21</font></i>: 
 <a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>;<br/>
<a NAME="E5:21"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:21_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp;  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E2:21_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span> &amp;  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PB</font>,<font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E3:21_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xboolean.html#K3">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11">BVFUNC_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1"/>
( <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>case </b></a><a NAME="E1:21_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:21_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><b>case </b></a><a NAME="E1:21_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_1_2_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; not <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E2:21_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:21_1_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E4:21_1_1_1_2"/><b>then </b>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span> holds <br/><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , <a class="ref" href="bvfunc_1.html#D18">BVFUNC_1:def 18</a></i>;<br/><a NAME="E5:21_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K17">I_el</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D14">BVFUNC_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E7:21"/>
 <a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font> <a href="bvfunc_1.html#R1">'&lt;'</a>  <a href="bvfunc_2.html#K6">All</a> <font color="Maroon">u</font>,<font color="Maroon">PB</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#T19">BVFUNC_1:19</a></i>;<br/>


</div>
