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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> ;<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> ;<br/><b>let </b><font color="Maroon" title="c3">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="E1:54"/>
for <font color="Olive" title="b1">F</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA</font> holds <br/>for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">F</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:54_1"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:54_1"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:54_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:54_1"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<a NAME="E5:54_1"/><i><font color="Green" title="E28">A3</font></i>: 
(  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> or  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">PA</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:54_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E6:54_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">PA</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E7:54_1"/>
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c3">PA</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E8:54_1"/><b>then </b>
 <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c3">PA</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E2:54_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:54_1"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
 <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c3">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:54_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1"/>
( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">PA</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  or  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">PA</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">PA</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:54_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D19" target="_self" title="BVFUNC_1:def.19">Def19</a></i>;<br/><b>hence </b><a NAME="E3:54_1_1_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:54_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:54_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:54_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="bvfunc_1.html#E9:54_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:54_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="bvfunc_1.html#D19" target="_self" title="BVFUNC_1:def.19">Def19</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2"/><i><font color="Green" title="E31">A7</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">PA</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ;<br/><div class="add"><a NAME="E2:54_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D19" target="_self" title="BVFUNC_1:def.19">Def19</a></i>;<br/><b>hence </b><a NAME="E3:54_1_1_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:54_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:54_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:54_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="bvfunc_1.html#E9:54_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:54_1_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="bvfunc_1.html#D19" target="_self" title="BVFUNC_1:def.19">Def19</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:54"/>
 <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a> <font color="Maroon" title="c3">PA</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D18" target="_self" title="BVFUNC_1:def.18">Def18</a></i>;<br/>


</div>
