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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">d</font> be  <a href="funct_1.html#V3">constant</a>  <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/>



<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E20</font></i>: 
( 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">d</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>  or 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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_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">d</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>  ) &amp; 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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or ( ( 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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; 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">d</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>  ) or ( ( 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">d</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>  ) &amp; ( 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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1"><b>case </b></a><a NAME="E1:41_1_1_1_1"/><i><font color="Green">E29</font></i>: 
( ( 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">d</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>  ) &amp; 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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_1_1_1_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a>, <a class="txt" href="bvfunc_1.html#E1:41"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_2"><b>case </b></a><a NAME="E1:41_1_1_1_2"/><i><font color="Green">E31</font></i>: 
( ( 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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; 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">d</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/><div class="add"><a NAME="E2:41_1_1_1_2"/><b>then </b>
<font color="Maroon">d</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/>;<br/><b>hence </b><a NAME="E3:41_1_1_1_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a>, <a class="txt" href="bvfunc_1.html#E1:41"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_3"><b>case </b></a><a NAME="E1:41_1_1_1_3"/><i><font color="Green">E43</font></i>: 
( ( 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">d</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>  ) &amp; ( 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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 ;<br/><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><a NAME="E2:41_1_1_1_3"/>
<font color="Maroon">d</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> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_1_1_1_3"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:41_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E44</font></i>: 
( not <!--HUMANRECFAILED-->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">d</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>  &amp; not <!--HUMANRECFAILED-->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">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2_1_1"/>
( <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font> or <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_1_1"><b>case </b></a><a NAME="E1:41_2_1_1_1"/>
<font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_2_1_1_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_1_2"><b>case </b></a><a NAME="E1:41_2_1_1_2"/>
<font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_2_1_1_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:41_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:41"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> )
 <b>by </b><i/>;<br/>


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