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





<div><i><font color="Green" title="E23">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E24">A2</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> holds  <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#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:46_1"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</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="txt" href="bvfunc_1.html#E1:46_1"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/><a NAME="E3:46_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E4:46_1"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E5:46_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <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="txt" href="bvfunc_1.html#E4:46_1"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:46_1"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D3" target="_self" title="BVFUNC_1:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2"/>
not 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> holds  <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#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:46_2"/><b>then </b>
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">I_el</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></i>;<br/><a NAME="E3:46_2"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</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#D14" target="_self" title="BVFUNC_1:def.14">Def14</a></i>;<br/><a NAME="E4:46_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <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="txt" href="bvfunc_1.html#E3:46_2"><i><font color="Green" title="E24">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:46_2"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D3" target="_self" title="BVFUNC_1:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:46"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:46"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>


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