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<span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font> iff  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E17">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon" title="c2">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:16_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> </span><br/><a NAME="E2:16_1_1"/><span class="kw">then </span>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="cqc_the3.html#D2" target="_self" title="CQC_THE3:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_1"/>
<font color="Maroon" title="c2">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#D9" title="CQC_THE1:def.9">CQC_THE1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:16_1"/>
 <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#D3" target="_self" title="CQC_THE3:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <font color="Maroon" title="c1">X</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_2_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">p</font></span><br/><a NAME="E2:16_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="cqc_the3.html#D3" target="_self" title="CQC_THE3:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_2_1"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#D9" title="CQC_THE1:def.9">CQC_THE1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:16_2"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#D2" target="_self" title="CQC_THE3:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:16"/>
(  <a href="cqc_the3.html#R3" title="CQC_THE3:pred.3">|-</a> <font color="Maroon" title="c1">X</font> iff  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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