<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:52_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E3:52_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:52_1"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E4:52_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:52_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E5:52_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#D16">MARGREL1:def 16</a></i>;<br/><a NAME="E6:52_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K1" target="_self">'imp'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E5:52_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/><b>hence </b><a NAME="E7:52_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:52_2"/><b>then </b>
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E3:52_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E4:52_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K1" target="_self">'imp'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:52_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/><b>hence </b><a NAME="E5:52_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:52"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
