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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ] means ( <font color="Maroon">a<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a> or <font color="Maroon">a<sub>1</sub></font> is <a href="hilbert2.html#V2" target="_self">conditional</a> or <font color="Maroon">a<sub>1</sub></font> is <a href="hilbert2.html#V3" target="_self">simple</a> or <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  );<br/>
<a NAME="E1:18"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
;<br/>
<a NAME="E2:18"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="ref" href="hilbert2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">b<sub>2</sub></font>] )
 
<b>by </b><i><a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:18"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" target="_self">HILBERT2:sch 2</a>(<a class="txt" href="hilbert2.html#E1:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="hilbert2.html#E2:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="hilbert2.html#E3:18"><i><font color="Green">E15</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:18"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V2" target="_self">conditional</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V3" target="_self">simple</a> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  )
 ;<br/>


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