<?xml version="1.0"?>
<div class="add">

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



<b>assume </b><a NAME="E1:47"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K6">Leaves</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>

<a NAME="E2:47"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="trees_1.html#K3">Leaves</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D10">TREES_2:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:47"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E5:47"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:47"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E8:47"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">q</font></span>)</span> <a href="trees_2.html#K7">|</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:47"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">p</font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:47_1"/>
 <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="trees_4.html#K10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="hilbert2.html#E8:47"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:47_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="trees_9.html#T6">TREES_9:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:47"/>
not <font color="Maroon">p</font> is <a href="hilbert2.html#V2" target="_self">conditional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_2"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V2" target="_self">conditional</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:47_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:47_2"/>
 <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="trees_4.html#K10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:47_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="trees_9.html#T6">TREES_9:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:47"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  or <font color="Maroon">p</font> is <a href="hilbert2.html#V3" target="_self">simple</a> )
 <b>by </b><i><a class="ref" href="hilbert2.html#T2" target="_self">Th2</a>, </i>;<br/>


</div>
