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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>, <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/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/>
<a NAME="E1:52"/><i><font color="Green">E51</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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) )
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:52"/><i><font color="Green">E52</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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <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">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <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">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>(<a class="txt" href="anproj_2.html#E1:52"><i><font color="Green">E51</font></i></a>);<br/></i>
<a NAME="E4:52"/><i><font color="Green">E53</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">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) )
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:52"/><i><font color="Green">E54</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">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>7</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">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>7</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">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>(<a class="txt" href="anproj_2.html#E4:52"><i><font color="Green">E53</font></i></a>);<br/></i>
<a NAME="E7:52"/><i><font color="Green">E55</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">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) &amp; (  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) )
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:52"/><i><font color="Green">E56</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">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <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">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <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">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>(<a class="txt" href="anproj_2.html#E7:52"><i><font color="Green">E55</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E10:52"/><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">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) &amp; (  not <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) )
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:52"/><i><font color="Green">E58</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">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>9</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">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>9</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">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>(<a class="txt" href="anproj_2.html#E10:52"><i><font color="Green">E57</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <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="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E14:52"/>
( ( 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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> implies <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> 1 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> implies <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> 0 ) ) ) &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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) ) ) &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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies <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> 1 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> implies <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> 0 ) ) ) &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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) ) ) )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:52"><i><font color="Green">E52</font></i></a>, <a class="txt" href="anproj_2.html#E6:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="anproj_2.html#E9:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="anproj_2.html#E12:52"><i><font color="Green">E58</font></i></a></i>;<br/>


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