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

<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="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ;<br/>

<div><i><font color="Green" title="E17">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E18">A2</font></i>: 
 <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ;<br/><b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E2:18_1_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> is <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:18_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="cqc_the3.html#D3" target="_self" title="CQC_THE3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 <b>by </b><i><a class="ref" href="cqc_the1.html#D11" title="CQC_THE1:def.11">CQC_THE1:def 11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 ;<br/><a NAME="E2:18_2"/><b>then </b>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">p</font> is <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the1.html#D11" title="CQC_THE1:def.11">CQC_THE1:def 11</a></i>;<br/><b>hence </b><a NAME="E3:18_2"/>
 <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D3" target="_self" title="CQC_THE3:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
(  <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font> iff <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a>  )
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>


</div>
