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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ,   <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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $3 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  &amp; $4 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  implies  ex <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $4 &amp; $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> $2</span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> ) ) &amp; ( (  not $3 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or  not $4 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  ) implies $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ,   <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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $3 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  &amp; $4 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  implies  ex <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $4 &amp; $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> $2</span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> ) ) &amp; ( (  not $3 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or  not $4 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  ) implies $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<a NAME="E1:40_1_1"/><i><font color="Green" title="E31">A1</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b5">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>]</span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">c</font>]</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_1_1"/>
( ( <font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  &amp; <font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  ) or  not <font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or  not <font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:40_1_1_1_1_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 <b>and </b><br/><a NAME="E2:40_1_1_1_1_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">c</font>]</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c3">a</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  <b>by </b><i><a class="txt" href="#E1:40_1_1_1_1_1"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">q'</font> = <font color="Maroon" title="c4">b</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  <b>by </b><i><a class="txt" href="#E2:40_1_1_1_1_1"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/><b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">q'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">q'</font>]</span><br/><b>thus </b><a NAME="E5:40_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">q'</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_2"/>
(  not <font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or  not <font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">c</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">c</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:40_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b5">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">d</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">d</font>]</span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">d</font>]</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_2_1"/>
( ( <font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  &amp; <font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  ) or  not <font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or  not <font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:40_1_1_2_1_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 <b>and </b><br/><a NAME="E2:40_1_1_2_1_1"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">d</font>]</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c3">a</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  <b>by </b><i><a class="txt" href="#E1:40_1_1_2_1_1"><i><font color="Green" title="E33">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">q'</font> = <font color="Maroon" title="c4">b</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  <b>by </b><i><a class="txt" href="#E2:40_1_1_2_1_1"><i><font color="Green" title="E34">A6</font></i></a></i>;<br/><b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">q'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">q'</font>]</span><br/><b>thus </b><a NAME="E5:40_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">q'</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_2_1_2"/>
(  not <font color="Maroon" title="c3">a</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or  not <font color="Maroon" title="c4">b</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">d</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">d</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:40_1_1"/><i><font color="Green" title="E33">A7</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">d</font>] holds <br/><font color="Olive" title="b5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">d</font>
 
;<br/>
<a NAME="E4:40_1_1"/><i><font color="Green" title="E34">A8</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">d</font>] holds <br/><font color="Olive" title="b5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">d</font>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c1">M</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> <b> such that </b><br/><a NAME="E6:40_1_1"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> 
 <b>and </b><br/><a NAME="E7:40_1_1"/><i><font color="Green" title="E36">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>and </b><br/><a NAME="E8:40_1_1"/><i><font color="Green" title="E37">A11</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>  holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span>] )
 <b>from </b><i><a class="ref" href="hilbert2.html#S3" target="_self" title="HILBERT2:sch.3">HILBERT2:sch 3</a>(<a class="txt" href="#E1:40_1_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E2:40_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E3:40_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="#E4:40_1_1"><i><font color="Green" title="E34">A8</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font> ) ) )</span><br/>

<b>thus </b><a NAME="E9:40_1_1"/>
<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> 
 <b>by </b><i><a class="txt" href="#E6:40_1_1"><i><font color="Green" title="E35">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font> ) ) )</span><br/>

<b>thus </b><a NAME="E10:40_1_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:40_1_1"><i><font color="Green" title="E36">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font> )</span><br/>

<b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> )</span><br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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>means </b><font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ;<br/>
<a NAME="E11:40_1_1"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> ]
 
<b>by </b><i><a class="txt" href="#E6:40_1_1"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/>
<a NAME="E12:40_1_1"/><i><font color="Green" title="E39">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font>]</span><br/>

<a NAME="E1:40_1_1_3"/>
<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:40_1_1"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:40_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:40_1_1"/><i><font color="Green" title="E40">A14</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">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>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">s</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">s</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">r</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">s</font>] implies ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">s</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font>] ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40_1_1_4"/><i><font color="Green" title="E41">A15</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 <b>and </b><br/><a NAME="E2:40_1_1_4"/><i><font color="Green" title="E42">A16</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">s</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font>] )</span><br/>

<a NAME="E3:40_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">s</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="#E8:40_1_1"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:40_1_1_4"/>
<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">s</font></span>)</span> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 <b>by </b><i><a class="txt" href="#E1:40_1_1_4"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="#E2:40_1_1_4"><i><font color="Green" title="E42">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font>]</span><br/>

<a NAME="E5:40_1_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="#E8:40_1_1"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:40_1_1_4"/>
<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font></span>)</span> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 <b>by </b><i><a class="txt" href="#E1:40_1_1_4"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="#E2:40_1_1_4"><i><font color="Green" title="E42">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:40_1_1"/><i><font color="Green" title="E41">A17</font></i>: 
for <font color="Olive" title="b1">p</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>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" target="_self" title="HILBERT2:sch.2">HILBERT2:sch 2</a>(<a class="txt" href="#E11:40_1_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="#E12:40_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="#E13:40_1_1"><i><font color="Green" title="E40">A14</font></i></a>);<br/></i>
<a NAME="E15:40_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>] )
 
<b>by </b><i><a class="txt" href="#E8:40_1_1"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:40_1_1"/>
 ex <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> &amp; <font color="Maroon" title="c1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font> )
 <b>by </b><i><a class="txt" href="#E14:40_1_1"><i><font color="Green" title="E41">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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