<?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="funct_1.html#V3">constant</a>  <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/>





<a NAME="E1:49"/><i><font color="Green">E26</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E2:49"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green">E28</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E4:49"/><i><font color="Green">E29</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:49"/><i><font color="Green">E30</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</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="E2:49_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E3:49_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>
<b>hence </b><a NAME="E4:49_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D14" target="_self">Def14</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:49"/><i><font color="Green">E31</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:49"/><i><font color="Green">E32</font></i>: 
(  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E10:49"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E5:49"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bvfunc_1.html#E7:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bvfunc_1.html#E9:49"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E11:49"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="txt" href="bvfunc_1.html#E7:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bvfunc_1.html#E9:49"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E12:49"/><i><font color="Green">E34</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_2"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</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#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</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/>
<b>hence </b><a NAME="E3:49_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:49"/><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:49"/><i><font color="Green">E36</font></i>: 
(  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E17:49"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="bvfunc_1.html#E14:49"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:49"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E18:49"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="txt" href="bvfunc_1.html#E14:49"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E19:49"/><i><font color="Green">E38</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_3"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</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#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_3"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E3:49_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</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#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</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#K8">TRUE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:49_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K7">FALSE</a>  <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#E1:49_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="margrel1.html#D16">MARGREL1:def 16</a></i>;<br/>
<a NAME="E5:49_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/>
<b>hence </b><a NAME="E6:49_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D13" target="_self">Def13</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:49"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E23:49"/><i><font color="Green">E40</font></i>: 
(  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E24:49"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E19:49"><i><font color="Green">E38</font></i></a>, <a class="txt" href="bvfunc_1.html#E21:49"><i><font color="Green">E39</font></i></a>, <a class="txt" href="bvfunc_1.html#E23:49"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E25:49"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="txt" href="bvfunc_1.html#E21:49"><i><font color="Green">E39</font></i></a>, <a class="txt" href="bvfunc_1.html#E23:49"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E26:49"/><i><font color="Green">E42</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_4"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E2:49_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <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="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E3:49_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <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="E4:49_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>
<b>hence </b><a NAME="E5:49_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D14" target="_self">Def14</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E28:49"/><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E30:49"/><i><font color="Green">E44</font></i>: 
(  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E31:49"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E26:49"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bvfunc_1.html#E28:49"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bvfunc_1.html#E30:49"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E32:49"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="txt" href="bvfunc_1.html#E28:49"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bvfunc_1.html#E30:49"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_5"/><i><font color="Green">E47</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>3</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#K8">TRUE</a>  or 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>3</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_5_1_1"/>
( ( ( 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>3</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#K8">TRUE</a>  ) &amp; 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>3</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>  ) or ( ( 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>3</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>  ) &amp; 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>3</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#K8">TRUE</a>  ) or ( ( 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>3</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#K8">TRUE</a>  ) &amp; ( 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>3</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>  ) ) )
 </span><b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5"><i><font color="Green">E47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_1"><b>case </b></a><a NAME="E1:49_5_1_1_1"/><i><font color="Green">E48</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>3</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#K8">TRUE</a>  ) &amp; 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>3</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/><div class="add"><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_5_1_1_1_1"/><i><font color="Green">E50</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#K8">TRUE</a>  or 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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_5_1_1_1_1_1_1"/>
( ( ( 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#K8">TRUE</a>  ) &amp; 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>  ) or ( ( 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>  ) &amp; 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#K8">TRUE</a>  ) or ( ( 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>  ) &amp; ( 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#K8">TRUE</a>  ) ) )
 </span><b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:49_5_1_1_1_1_1_1_1"/><i><font color="Green">E51</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#K8">TRUE</a>  ) &amp; 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/><div class="add"><a NAME="E2:49_5_1_1_1_1_1_1_1"/><i><font color="Green">E52</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E2:49_5_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:49_5_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_5_1_1_1_1_1_1_1"/><i><font color="Green">E53</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_1_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E2:49_5_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:49_5_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_5_1_1_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E5:49_5_1_1_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E6:49_5_1_1_1_1_1_1_1"/><i><font color="Green">E56</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#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E7:49_5_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E11:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:49_5_1_1_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><a NAME="E8:49_5_1_1_1_1_1_1_1"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:49"><i><font color="Green">E28</font></i></a>, <a class="txt" href="bvfunc_1.html#E11:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:49_5_1_1_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:49_5_1_1_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:49_5_1_1_1_1_1_1_2"/><i><font color="Green">E51</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>  ) &amp; 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#K8">TRUE</a>  )
 ;<br/><div class="add"><a NAME="E2:49_5_1_1_1_1_1_1_2"/><i><font color="Green">E52</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_1_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_1_1_1_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_1_1_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E4:49_5_1_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K7">FALSE</a>  <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#E1:49_5_1_1_1_1_1_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_1_1_1_1_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="margrel1.html#D16">MARGREL1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E5:49_5_1_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_5_1_1_1_1_1_1_2"/><i><font color="Green">E53</font></i>: 
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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_5_1_1_1_1_1_1_2_2_1"/><i><font color="Green">E54</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:49_5_1_1_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_2_2_1"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_5_1_1_1_1_1_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49_5_1_1_1_1_1_1_2"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:49_5_1_1_1_1_1_1_2_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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:49_5_1_1_1_1_1_1_2"/><i><font color="Green">E54</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_1_1_1_2_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E2:49_5_1_1_1_1_1_1_2_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_1_1_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_1_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <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#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_2_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:49_5_1_1_1_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:49_5_1_1_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E6:49_5_1_1_1_1_1_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E7:49_5_1_1_1_1_1_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E8:49_5_1_1_1_1_1_1_2"/>
 <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="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E9:49_5_1_1_1_1_1_1_2"/><b>then </b>
<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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E18:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:49_5_1_1_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E10:49_5_1_1_1_1_1_1_2"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E25:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:49_5_1_1_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:49_5_1_1_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="bvfunc_1.html#E7:49_5_1_1_1_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:49_5_1_1_1_1_1_1_3"/><i><font color="Green">E51</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>  ) &amp; ( 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#K8">TRUE</a>  ) )
 ;<br/><div class="add"><a NAME="E2:49_5_1_1_1_1_1_1_3"/><i><font color="Green">E52</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_1_1_1_3_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_1_1_1_1_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_3"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:49_5_1_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_5_1_1_1_1_1_1_3"/><i><font color="Green">E53</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_1_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_1_1_1_3_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <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>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_1_1_1_1_3_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:49_5_1_1_1_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_3_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_5_1_1_1_1_1_1_3"/><i><font color="Green">E54</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E5:49_5_1_1_1_1_1_1_3"/><i><font color="Green">E55</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E6:49_5_1_1_1_1_1_1_3"/>
 <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="txt" href="bvfunc_1.html#E1:49_5_1_1_1_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><a NAME="E7:49_5_1_1_1_1_1_1_3"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bvfunc_1.html#E11:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_1_1_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:49_5_1_1_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_1_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:49_5_1_1_1_1_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_5_1_1_1_1"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_5_1_1_1_2"/><i><font color="Green">E50</font></i>: 
( not <!--HUMANRECFAILED-->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#K8">TRUE</a>  &amp; not <!--HUMANRECFAILED-->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="E2:49_5_1_1_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E3:49_5_1_1_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E4:49_5_1_1_1_2"/><i><font color="Green">E53</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#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E5:49_5_1_1_1_2"/><i><font color="Green">E54</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_2_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E3:49_5_1_1_1_2_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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:49_5_1_1_1_2_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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49"><i><font color="Green">E26</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E5:49_5_1_1_1_2_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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E6:49_5_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="bvfunc_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:49_5_1_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:49_5_1_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E10:49_5_1_1_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E5:49_5_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E7:49_5_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="bvfunc_1.html#E9:49_5_1_1_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E11:49_5_1_1_1_2"/><i><font color="Green">E58</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_1_2_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <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>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_1_2_2"/>
<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>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E3:49_5_1_1_1_2_2"/><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>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:49_5_1_1_1_2_2"/><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>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49"><i><font color="Green">E26</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E5:49_5_1_1_1_2_2"/><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>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E6:49_5_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_1_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self">Def14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:49_5_1_1_1_2"/>
<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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E11:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E13:49_5_1_1_1_2"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E25:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:49_5_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E7:49_5_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="bvfunc_1.html#E9:49_5_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="bvfunc_1.html#E10:49_5_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="bvfunc_1.html#E11:49_5_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:49_5_1_1_1"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49_5_1_1_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_2"><b>case </b></a><a NAME="E1:49_5_1_1_2"/><i><font color="Green">E48</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>3</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>  ) &amp; 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>3</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#K8">TRUE</a>  )
 ;<br/><div class="add"><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_5_1_1_2_1"/><i><font color="Green">E50</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#K8">TRUE</a>  or 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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_5_1_1_2_1_1_1"/>
( ( ( 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#K8">TRUE</a>  ) &amp; 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>  ) or ( ( 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>  ) &amp; 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#K8">TRUE</a>  ) or ( ( 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>  ) &amp; ( 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#K8">TRUE</a>  ) ) )
 </span><b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:49_5_1_1_2_1_1_1_1"/><i><font color="Green">E51</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#K8">TRUE</a>  ) &amp; 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/><div class="add"><a NAME="E2:49_5_1_1_2_1_1_1_1"/><i><font color="Green">E52</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E2:49_5_1_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <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#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:49_5_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_5_1_1_2_1_1_1_1"/><i><font color="Green">E53</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_1_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E2:49_5_1_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K8">TRUE</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#E1:49_5_1_1_2_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E4:49_5_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K7">FALSE</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/>
<b>hence </b><a NAME="E5:49_5_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_5_1_1_2_1_1_1_1"/><i><font color="Green">E54</font></i>: 
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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_5_1_1_2_1_1_1_1_3_1"/><i><font color="Green">E55</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:49_5_1_1_2_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_1_3_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_5_1_1_2_1_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:49_5_1_1_2_1_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:49_5_1_1_2_1_1_1_1_3"/>
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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E5:49_5_1_1_2_1_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E6:49_5_1_1_2_1_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E7:49_5_1_1_2_1_1_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E8:49_5_1_1_2_1_1_1_1"/>
 <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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E9:49_5_1_1_2_1_1_1_1"/><b>then </b>
<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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E25:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:49_5_1_1_2_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E10:49_5_1_1_2_1_1_1_1"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_2_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:49_5_1_1_2_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="bvfunc_1.html#E7:49_5_1_1_2_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:49_5_1_1_2_1_1_1_2"/><i><font color="Green">E51</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>  ) &amp; 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#K8">TRUE</a>  )
 ;<br/><div class="add"><a NAME="E2:49_5_1_1_2_1_1_1_2"/><i><font color="Green">E52</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_1_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E2:49_5_1_1_2_1_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2_1_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <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#E1:49_5_1_1_2_1_1_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E4:49_5_1_1_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <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/>
<b>hence </b><a NAME="E5:49_5_1_1_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_5_1_1_2_1_1_1_2"/><i><font color="Green">E53</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_1_1_1_2_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <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>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_2_1_1_1_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_2_1_1_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2_1_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E4:49_5_1_1_2_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2_1_1_1_2_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="margrel1.html#D16">MARGREL1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E5:49_5_1_1_2_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_5_1_1_2_1_1_1_2"/><i><font color="Green">E54</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E5:49_5_1_1_2_1_1_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E6:49_5_1_1_2_1_1_1_2"/>
 <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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><a NAME="E7:49_5_1_1_2_1_1_1_2"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bvfunc_1.html#E32:49"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:49_5_1_1_2_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_2_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:49_5_1_1_2_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:49_5_1_1_2_1_1_1_3"/><i><font color="Green">E51</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>  ) &amp; ( 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#K8">TRUE</a>  ) )
 ;<br/><div class="add"><a NAME="E2:49_5_1_1_2_1_1_1_3"/><i><font color="Green">E52</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_1_1_1_3_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_2_1_1_1_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_3"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:49_5_1_1_2_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_5_1_1_2_1_1_1_3"/><i><font color="Green">E53</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_1_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_1_1_1_3_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <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>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E2:49_5_1_1_2_1_1_1_3_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:49_5_1_1_2_1_1_1_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_2_1_1_1_3"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E4:49_5_1_1_2_1_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_3_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2_1_1_1_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="margrel1.html#D16">MARGREL1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E5:49_5_1_1_2_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_5_1_1_2_1_1_1_3"/><i><font color="Green">E54</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E5:49_5_1_1_2_1_1_1_3"/><i><font color="Green">E55</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E6:49_5_1_1_2_1_1_1_3"/>
 <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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E7:49_5_1_1_2_1_1_1_3"/><b>then </b>
<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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E32:49"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_2_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="bvfunc_1.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E8:49_5_1_1_2_1_1_1_3"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2_1_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:49_5_1_1_2_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49_5_1_1_2_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:49_5_1_1_2_1_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_5_1_1_2_1"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_5_1_1_2_2"/><i><font color="Green">E50</font></i>: 
( not <!--HUMANRECFAILED-->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#K8">TRUE</a>  &amp; not <!--HUMANRECFAILED-->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="E2:49_5_1_1_2_2"/><i><font color="Green">E51</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E3:49_5_1_1_2_2"/><i><font color="Green">E52</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E2:49_5_1_1_2_2_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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49"><i><font color="Green">E26</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:49_5_1_1_2_2_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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:49_5_1_1_2_2_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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:49_5_1_1_2_2_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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <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>12</sub></font></span>)</span> <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/>
<b>hence </b><a NAME="E6:49_5_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#D11" target="_self">Def11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_5_1_1_2_2"/><i><font color="Green">E53</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_5_1_1_2_2_2"/>
<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>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E2:49_5_1_1_2_2_2"/><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>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:49_5_1_1_2_2_2"/><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>12</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49"><i><font color="Green">E26</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bvfunc_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E4:49_5_1_1_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D11" target="_self">Def11</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:49_5_1_1_2_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:49_5_1_1_2_2"/><i><font color="Green">E55</font></i>: 
(  <a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E9:49_5_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E4:49_5_1_1_2_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:49_5_1_1_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E8:49_5_1_1_2_2"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E10:49_5_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E6:49_5_1_1_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bvfunc_1.html#E8:49_5_1_1_2_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E11:49_5_1_1_2_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="valuat_1.html#K5">'not'</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>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E12:49_5_1_1_2_2"/><i><font color="Green">E57</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E13:49_5_1_1_2_2"/><i><font color="Green">E58</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E14:49_5_1_1_2_2"/>
 <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="txt" href="bvfunc_1.html#E1:49_5_1_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><a NAME="E15:49_5_1_1_2_2"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bvfunc_1.html#E4:49"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bvfunc_1.html#E25:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="bvfunc_1.html#E32:49"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bvfunc_1.html#E3:49_5_1_1_2_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E11:49_5_1_1_2_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="bvfunc_1.html#E12:49_5_1_1_2_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="bvfunc_1.html#E13:49_5_1_1_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a>, <a class="ref" href="bvfunc_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:49_5_1_1_2"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:49_5_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_5_1_1_3"><b>case </b></a><a NAME="E1:49_5_1_1_3"/><i><font color="Green">E48</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>3</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#K8">TRUE</a>  ) &amp; ( 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>3</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/><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:49_5_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</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#E1:49_5_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_5_1_1_3"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_5_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_5"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_6"/><i><font color="Green">E47</font></i>: 
( not <!--HUMANRECFAILED-->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>3</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#K8">TRUE</a>  &amp; not <!--HUMANRECFAILED-->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>3</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_6_1_1"/>
( <font color="Maroon">c<sub>3</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> or <font color="Maroon">c<sub>3</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> )
 </span><b>by </b><i><a class="ref" href="bvfunc_1.html#T24" target="_self">Th24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_6_1_1_1"><b>case </b></a><a NAME="E1:49_6_1_1_1"/>
<font color="Maroon">c<sub>3</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_6_1_1_1"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_6_1_1_2"><b>case </b></a><a NAME="E1:49_6_1_1_2"/>
<font color="Maroon">c<sub>3</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_6_1_1_2"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:49_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_6"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E35:49"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K14" target="_self">'imp'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E33:49"><i><font color="Green">E46</font></i></a></i>;<br/>


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