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<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>





<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:45_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E3:45_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:45_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2"/>
not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:45_2"/><b>then </b>
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:45_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E4:45_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K8">TRUE</a>  <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/><a NAME="E5:45_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K6">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:45_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">'&lt;'</a> <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


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