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<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V1" title="HILBERT2:attr.1">conjunctive</a> or <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V2" title="HILBERT2:attr.2">conditional</a> or <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a> or <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ] <b>means </b>( $1 is <a href="hilbert2.html#V1" title="HILBERT2:attr.1">conjunctive</a> or $1 is <a href="hilbert2.html#V2" title="HILBERT2:attr.2">conditional</a> or $1 is <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a> or $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  );<br/>
<a NAME="E1:18"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> ]
 
;<br/>
<a NAME="E2:18"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font>]
 
<b>by </b><i><a class="ref" href="hilbert2.html#D8" target="_self" title="HILBERT2:def.8">Def8</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green" title="E15">A3</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">s</font>] )
 
<b>by </b><i><a class="ref" href="hilbert2.html#D6" target="_self" title="HILBERT2:def.6">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self" title="HILBERT2:def.7">Def7</a></i>;<br/>
<a NAME="E4:18"/>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" target="_self" title="HILBERT2:sch.2">HILBERT2:sch 2</a>(<a class="txt" href="hilbert2.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="hilbert2.html#E2:18"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="hilbert2.html#E3:18"><i><font color="Green" title="E15">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:18"/>
( <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V1" title="HILBERT2:attr.1">conjunctive</a> or <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V2" title="HILBERT2:attr.2">conditional</a> or <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a> or <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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