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<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:21"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V2" target="_self">conditional</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V3" target="_self">simple</a> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  )
 </span><b>by </b><i><a class="ref" href="hilbert2.html#T9" target="_self">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V2" target="_self">conditional</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E3:21_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D7" target="_self">Def7</a></i>;<br/><font color="Maroon">c<sub>1</sub></font> = 
<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">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="hilbert2.html#E3:21_1_2"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<span class="p1"><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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><b>hence </b><a NAME="E5:21_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_3"><b>suppose </b></a><a NAME="E1:21_1_3"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V3" target="_self">simple</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:21_1_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E4:21_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:21_1_3"><i><font color="Green">E17</font></i></a></i>;<br/><b>then </b>2 <a href="nat_1.html#K1">+</a> 0 = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="hilbert2.html#E1:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
2 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/><b>hence </b><a NAME="E6:21_1_3"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_4"><b>suppose </b></a><a NAME="E1:21_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_4"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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