<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a> holds <br/>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font>;<br/>
<b>consider </b><font color="Maroon" title="c1">D0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:8_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D0</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:8_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:8_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a> holds <br/><span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_1_1"/>
<font color="Maroon" title="c2">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a>
 ;<br/>

<a NAME="E2:8_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">D</font> is <a href="hilbert1.html#V1" title="HILBERT1:attr.1">with_VERUM</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D5" target="_self" title="HILBERT1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="ref" href="hilbert1.html#D1" target="_self" title="HILBERT1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">D0</font> = <font color="Maroon" title="c1">D0</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E3:8_1_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">D0</font>
;<br/>

<a NAME="E6:8_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c2">D0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:8_1_1_2"/>
(  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:8_1_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 
<b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E3:8_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="hilbert1.html#E4:8_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E8:8_1_1"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c2">D0</font> is <a href="hilbert1.html#V1" title="HILBERT1:attr.1">with_VERUM</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D1" target="_self" title="HILBERT1:def.1">Def1</a></i>;<br/>
<a NAME="E9:8_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  <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3"><b>proof </b></a><div class="add">

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

<b>set </b><font color="Maroon" title="c4">p</font> = 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">h</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E2:8_1_1_3"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:8_1_1_3"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a> holds <br/><span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_1_3_1"/>
<font color="Maroon" title="c6">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a>
 ;<br/>

<a NAME="E2:8_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c6">D</font> is <a href="hilbert1.html#V4" title="HILBERT1:attr.4">with_propositional_variables</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D5" target="_self" title="HILBERT1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_3_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="hilbert1.html#D4" target="_self" title="HILBERT1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1_1_3"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E2:8_1_1_3"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:8_1_1"/><b>then </b><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c2">D0</font> is <a href="hilbert1.html#V4" title="HILBERT1:attr.4">with_propositional_variables</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D4" target="_self" title="HILBERT1:def.4">Def4</a></i>;<br/>
<a NAME="E11:8_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:8_1_1_4"/><i><font color="Green" title="E13">A8</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> )
 ;<br/>

<a NAME="E2:8_1_1_4"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )
 
<b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E1:8_1_1_4"><i><font color="Green" title="E13">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c6">q'</font> = <font color="Maroon" title="c4">q</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:8_1_1_4"/><i><font color="Green" title="E14">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">p'</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E5:8_1_1_4"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_1_4_1"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c7">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a>
 ;<br/>

<a NAME="E2:8_1_1_4_1"/><b>then </b><i><font color="Green" title="E16">A11</font></i>: 
<font color="Maroon" title="c7">D</font> is <a href="hilbert1.html#V2" title="HILBERT1:attr.2">with_implication</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D5" target="_self" title="HILBERT1:def.5">Def5</a></i>;<br/>
<a NAME="E3:8_1_1_4_1"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> )
 
<b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E1:8_1_1_4"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="hilbert1.html#E1:8_1_1_4_1"><i><font color="Green" title="E15">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:8_1_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1_4_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="hilbert1.html#D2" target="_self" title="HILBERT1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:8_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E4:8_1_1_4"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:8_1_1"/><b>then </b><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c2">D0</font> is <a href="hilbert1.html#V2" title="HILBERT1:attr.2">with_implication</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D2" target="_self" title="HILBERT1:def.2">Def2</a></i>;<br/>
<a NAME="E13:8_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:8_1_1_5"/><i><font color="Green" title="E14">A13</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font> )
 ;<br/>

<a NAME="E2:8_1_1_5"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )
 
<b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E1:8_1_1_5"><i><font color="Green" title="E14">A13</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c6">q'</font> = <font color="Maroon" title="c4">q</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:8_1_1_5"/><i><font color="Green" title="E15">A14</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">p'</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E5:8_1_1_5"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_1_5_1"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c7">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a>
 ;<br/>

<a NAME="E2:8_1_1_5_1"/><b>then </b><i><font color="Green" title="E17">A16</font></i>: 
<font color="Maroon" title="c7">D</font> is <a href="hilbert1.html#V3" title="HILBERT1:attr.3">with_conjunction</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D5" target="_self" title="HILBERT1:def.5">Def5</a></i>;<br/>
<a NAME="E3:8_1_1_5_1"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> )
 
<b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E1:8_1_1_5"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="hilbert1.html#E1:8_1_1_5_1"><i><font color="Green" title="E16">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:8_1_1_5_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1_5_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="hilbert1.html#D3" target="_self" title="HILBERT1:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:8_1_1_5"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E4:8_1_1_5"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:8_1_1"/><b>then </b>
<font color="Maroon" title="c2">D0</font> is <a href="hilbert1.html#V3" title="HILBERT1:attr.3">with_conjunction</a>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D3" target="_self" title="HILBERT1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E15:8_1_1"/>
<font color="Maroon" title="c2">D0</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a>
 <b>by </b><i><a class="txt" href="hilbert1.html#E6:8_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="hilbert1.html#E8:8_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="hilbert1.html#E10:8_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="hilbert1.html#E12:8_1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="hilbert1.html#D5" target="_self" title="HILBERT1:def.5">Def5</a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E16:8_1_1"/><i><font color="Green" title="E14">A17</font></i>: 
<font color="Maroon" title="c3">D</font> is <a href="hilbert1.html#V5" title="HILBERT1:attr.5">HP-closed</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E17:8_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D0</font>
 ;<br/>

<b>hence </b><a NAME="E18:8_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:8_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="hilbert1.html#E16:8_1_1"><i><font color="Green" title="E14">A17</font></i></a></i>;<br/>


</div>
