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<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><a href="hilbert2.html#K2" title="HILBERT2:func.2">HP-Subformulae</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ;<br/>
<a NAME="E1:41_1_1"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> ]
 
<b>by </b><i><a class="ref" href="hilbert2.html#D9" target="_self" title="HILBERT2:def.9">Def9</a></i>;<br/>
<a NAME="E2:41_1_1"/><i><font color="Green" title="E33">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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c2">n</font>]</span><br/>

<a NAME="E1:41_1_1_1"/>
<a href="hilbert2.html#K2" title="HILBERT2:func.2">HP-Subformulae</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert2.html#D9" target="_self" title="HILBERT2:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E2:41_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:41_1_1"/><i><font color="Green" title="E34">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>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">s</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">s</font>] implies ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">s</font>] ) )</span><br/>



<a NAME="E1:41_1_1_2"/>
 ex <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="hilbert2.html#K2" title="HILBERT2:func.2">HP-Subformulae</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="hilbert2.html#K2" title="HILBERT2:func.2">HP-Subformulae</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> &amp; <a href="hilbert2.html#K2" title="HILBERT2:func.2">HP-Subformulae</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> &amp; <a href="hilbert2.html#K2" title="HILBERT2:func.2">HP-Subformulae</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> )
 
<b>by </b><i><a class="ref" href="hilbert2.html#D9" target="_self" title="HILBERT2:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E2:41_1_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">s</font>] implies ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">s</font>] ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:41_1_1"/>
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="#E1:41_1_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="#E2:41_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="#E3:41_1_1"><i><font color="Green" title="E34">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:41_1_1"/>
<a href="hilbert2.html#K2" title="HILBERT2:func.2">HP-Subformulae</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</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"> verum</span><br/>


</div>
