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<div class="add">

<b>let </b><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> ;<br/>



<b>assume </b><a NAME="E1:47"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K6" title="TREES_2:func.6">Leaves</a> <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>
 ;<br/>

<a NAME="E2:47"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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 href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D10" title="TREES_2:def.10">TREES_2:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:47"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <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>
 <b>and </b><br/><a NAME="E5:47"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:47"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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 href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">x</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <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> <b>by </b><i><a class="txt" href="hilbert2.html#E4:47"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<a NAME="E8:47"/><i><font color="Green" title="E40">A4</font></i>: 
<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 href="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E6:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="hilbert2.html#T35" target="_self" title="HILBERT2:th.35">Th35</a></i>;<br/>
<a NAME="E9:47"/><i><font color="Green" title="E41">A5</font></i>: 
not <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V1" title="HILBERT2:attr.1">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" title="c1">p</font> is <a href="hilbert2.html#V1" title="HILBERT2:attr.1">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D6" target="_self" title="HILBERT2:def.6">Def6</a></i>;<br/>
<a NAME="E4:47_1"/>
 <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c5">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c6">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:47_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="hilbert2.html#T32" target="_self" title="HILBERT2:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E5:47_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbert2.html#E5:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="hilbert2.html#E8:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="trees_9.html#T6" title="TREES_9:th.6">TREES_9:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:47"/>
not <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V2" title="HILBERT2:attr.2">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" title="c1">p</font> is <a href="hilbert2.html#V2" title="HILBERT2:attr.2">conditional</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E3:47_2"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D7" target="_self" title="HILBERT2:def.7">Def7</a></i>;<br/>
<a NAME="E4:47_2"/>
 <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c5">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c6">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:47_2"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="hilbert2.html#T33" target="_self" title="HILBERT2:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E5:47_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbert2.html#E5:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="hilbert2.html#E8:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="trees_9.html#T6" title="TREES_9:th.6">TREES_9:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:47"/>
( <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>  or <font color="Maroon" title="c1">p</font> is <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a> )
 <b>by </b><i><a class="txt" href="hilbert2.html#E9:47"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="hilbert2.html#T9" target="_self" title="HILBERT2:th.9">Th9</a></i>;<br/>


</div>
