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<div class="add">

<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> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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> ;<br/><b>let </b><font color="Maroon" title="c4">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>;<br/><b>let </b><font color="Maroon" title="c5">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>;<br/>







<a NAME="E1:16"/><i><font color="Green" title="E5">A1</font></i>: 
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#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'eqv'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'eqv'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> = 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="bvfunc_4.html#T7" target="_self" title="BVFUNC_4:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T42" title="BVFUNC_1:th.42">BVFUNC_1:42</a></i>
;<br/>
<b>hence </b><a NAME="E2:16_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'eqv'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">k3</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green" title="E6">A2</font></i>: 
(  <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'eqv'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k3</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">k3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">k4</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:16"/><i><font color="Green" title="E7">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k4</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">k4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">k4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:16"/>
( <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">k3</font> &amp; <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">k4</font> &amp; ( for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> holds <br/><font color="Maroon" title="c6">k3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> ) )
 
<b>by </b><i><a class="txt" href="bvfunc_4.html#E1:16"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="bvfunc_4.html#E3:16"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="bvfunc_4.html#E5:16"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:16"/>
 <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'eqv'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc_4.html#E3:16"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="bvfunc_4.html#E5:16"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
