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<div class="add">

<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> ] <b>means </b>not <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">V</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>;<br/>
<a NAME="E1:37_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> ]
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E2:37_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</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="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:37_1_1_1"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <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="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E2:37_1_1_1"/>
not <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <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="c3">n</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:37_1_1"/><i><font color="Green" title="E29">A3</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>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/>( <font color="Maroon">S<sub>1</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>1</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="37_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">r</font>, <font color="Maroon" title="c4">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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37_1_1_2"/><i><font color="Green" title="E30">A4</font></i>: 
not <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">r</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E2:37_1_1_2"/><i><font color="Green" title="E31">A5</font></i>: 
not <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<a NAME="E3:37_1_1_2"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1_1_2"/>
not <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c4">s</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:37_1_1_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="hilbert3.html#E2:37_1_1_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/>

<a NAME="E5:37_1_1_2"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:37_1_1_2"/>
not <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">s</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="hilbert3.html#E2:37_1_1_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:37_1_1"/>
for <font color="Olive" title="b1">r</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>1</sub></font>[<font color="Olive" title="b1">r</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" title="HILBERT2:sch.2">HILBERT2:sch 2</a>(<a class="txt" href="hilbert3.html#E1:37_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="hilbert3.html#E2:37_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="hilbert3.html#E3:37_1_1"><i><font color="Green" title="E29">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:37_1_1"/>
not  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>


</div>
