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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="pl_axiom.html#K1" title="PL_AXIOM:func.1">PL-WFF</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="pl_axiom.html#K1" title="PL_AXIOM:func.1">PL-WFF</a> holds  <font color="Olive" title="b1">F</font> <a href="pl_axiom.html#R6" title="PL_AXIOM:pred.6">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="pl_axiom.html#K1" title="PL_AXIOM:func.1">PL-WFF</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="pl_axiom.html#R6" title="PL_AXIOM:pred.6">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span><br/>



<a NAME="E1:93"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pl_axiom.html#K12" title="PL_AXIOM:func.12">PL_axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pl_axiom.html#D19" target="_self" onmouseover="rs('pl_axiom/D19')" onmouseout="rh()">withplax</a></span>;<br/>
<a NAME="E2:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="pl_axiom.html#R6" title="PL_AXIOM:pred.6">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pl_axiom.html#T53" target="_self" onmouseover="rs('pl_axiom/T53')" onmouseout="rh()">th42</a></span>;<br/>
<a NAME="E3:93"/>
<font color="Maroon" title="c2">F</font> <a href="pl_axiom.html#R6" title="PL_AXIOM:pred.6">|-</a> <span class="p1">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pl_axiom.html#T57" target="_self" onmouseover="rs('pl_axiom/T57')" onmouseout="rh()">thaa</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:93"/>
<font color="Maroon" title="c2">F</font> <a href="pl_axiom.html#R6" title="PL_AXIOM:pred.6">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pl_axiom.html#K6" title="PL_AXIOM:func.6">'not'</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="pl_axiom.html#K3" title="PL_AXIOM:func.3">=&gt;</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="pl_axiom.html#T54" target="_self" onmouseover="rs('pl_axiom/T54')" onmouseout="rh()">th43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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