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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c2">q</font></span>)</span>)</span><br/>



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 ex <font color="Olive" title="b1">p9</font>, <font color="Olive" title="b2">q9</font> being   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b1">p9</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="c1">p</font> &amp; <font color="Olive" title="b2">q9</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">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="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b1">p9</font>,<font color="Olive" title="b2">q9</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="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b1">p9</font>,<font color="Olive" title="b2">q9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert2.html#D9" target="_self" onmouseover="rs('hilbert2/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:43"/>
 <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c2">q</font></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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