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<b>thus </b><a NAME="E1:15_1_1"/>
( <font color="Maroon">D</font> is <a href="hilbert1.html#V2">with_implication</a> implies for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="hilbert1.html#D2">HILBERT1:def 2</a></i>;<br/>

<b>assume </b><a NAME="E2:15_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert1.html#D2">HILBERT1:def 2</a><br/>

<b>assume </b><a NAME="E3:15_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  ;<br/>
<a NAME="E5:15_1_1"/>
<font color="Maroon">p'</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:15_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>


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