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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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">  <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <span class="p1">(<span class="default"><a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">a</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <span class="p1">(<span class="default"><a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font></span>)</span> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font> )</span><br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <span class="p1">(<span class="default"><a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">t</font> = <font color="Maroon" title="c2">a</font> as    <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/>
<a NAME="E3:28"/>
for <font color="Olive" title="b1">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>   st <font color="Olive" title="b1">T</font> is <a href="hilbert1.html#V6" title="HILBERT1:attr.6">Hilbert_theory</a> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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" title="c4">T</font> is <a href="hilbert1.html#V6" title="HILBERT1:attr.6">Hilbert_theory</a> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T</font> implies <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c4">T</font> is <a href="hilbert1.html#V6" title="HILBERT1:attr.6">Hilbert_theory</a>
 <b>and </b><br/><a NAME="E2:28_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font></span><br/>

<a NAME="E3:28_1"/>
 <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="txt" href="hilbert1.html#E1:28_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="hilbert1.html#E2:28_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="hilbert1.html#T8" target="_self" title="HILBERT1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="hilbert1.html#E1:28_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="hilbert1.html#D11" target="_self" title="HILBERT1:def.11">Def11</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>
<b>hence </b><a NAME="E4:28"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K5" title="HILBERT1:func.5">CnPos</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="hilbert1.html#D11" target="_self" title="HILBERT1:def.11">Def11</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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