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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span><br/> for <font color="Olive" title="b2">a</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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="b3">PA</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Olive" title="b3">PA</font>,<font color="Olive" title="b1">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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">PA</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Olive" title="b2">PA</font>,<font color="Maroon" title="c2">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">PA</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Olive" title="b1">PA</font>,<font color="Maroon" title="c2">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">PA</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span><br/>







<a NAME="E1:21"/>
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/>

<a NAME="E1:21_1"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c5">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D14" title="BVFUNC_1:def.14">BVFUNC_1:def 14</a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:21_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:21"/>
 <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D14" title="BVFUNC_1:def.14">BVFUNC_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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