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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <br/> for <font color="Olive" title="b2">G</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span><br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b2">G</font> is <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a> holds <br/> <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b2">G</font></span>)</span>,<font color="Olive" title="b4">B</font>,<font color="Olive" title="b2">G</font> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">B</font>,<font color="Olive" title="b2">G</font></span>)</span>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b2">G</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</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">G</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">G</font> is <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a> holds <br/> <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">G</font></span>)</span>,<font color="Olive" title="b3">B</font>,<font color="Olive" title="b1">G</font> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b3">B</font>,<font color="Olive" title="b1">G</font></span>)</span>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">G</font></span><br/><b>let </b><font color="Maroon" title="c3">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>; <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>, <font color="Olive" title="b2">B</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Maroon" title="c3">G</font> is <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a> holds <br/> <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Olive" title="b2">B</font>,<font color="Maroon" title="c3">G</font> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c3">G</font></span><br/><b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</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> is <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a> implies  <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font> )</span><br/>









<b>assume </b><a NAME="E1:6"/>
<font color="Maroon" title="c3">G</font> is <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span><br/>

<a NAME="E2:6"/><b>then </b>
 <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="ref" href="partit_2.html#T17" title="PARTIT_2:th.17">PARTIT_2:17</a></i>;<br/>
<b>hence </b><a NAME="E3:6"/>
 <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="bvfunc11.html#T8" target="_self" title="BVFUNC11:th.8">Th8</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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