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<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>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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/>



<a NAME="E1:6"/><i><font color="Green">E1</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"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="bvfunc_1.html#K8">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="bvfunc_1.html#K8">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7">BVFUNC_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11">BVFUNC_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11">BVFUNC_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T20">BINARITH:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T20">BINARITH:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T20">BINARITH:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T21">BINARITH:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T9">BINARITH:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11">BVFUNC_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E2:6_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="bvfunc_1.html#K8">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<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="E3:6"/><i><font color="Green">E2</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</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/>
<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="E5:6"/><i><font color="Green">E3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="bvfunc_1.html#K8">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</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/>
<a NAME="E6:6"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( 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>5</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>6</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_7.html#E1:6"><i><font color="Green">E1</font></i></a>, <a class="txt" href="bvfunc_7.html#E3:6"><i><font color="Green">E2</font></i></a>, <a class="txt" href="bvfunc_7.html#E5:6"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="bvfunc_1.html#K8">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc_7.html#E3:6"><i><font color="Green">E2</font></i></a>, <a class="txt" href="bvfunc_7.html#E5:6"><i><font color="Green">E3</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
