<?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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
( <font color="Maroon">p</font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a> or <font color="Maroon">p</font> is <a href="hilbert2.html#V2" target="_self">conditional</a> or <font color="Maroon">p</font> is <a href="hilbert2.html#V3" target="_self">simple</a> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 ;<br/><div class="add"><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="E3:45_1_1"/><i><font color="Green">E29</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="E4:45_1_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>, </i>;<br/><b>hence </b><a NAME="E5:45_1_1"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V2" target="_self">conditional</a>
 ;<br/><div class="add"><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="E3:45_1_2"/><i><font color="Green">E30</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="E4:45_1_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="E5:45_1_2"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_3"><b>suppose </b></a><a NAME="E1:45_1_3"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V3" target="_self">simple</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E4:45_1_3"/>
 <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="hilbert2.html#E3:45_1_3"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E5:45_1_3"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4"><b>suppose </b></a><a NAME="E1:45_1_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_4"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
