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<span class="kw">let </span><font color="Maroon" title="c4">Q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K8" title="LTLAXIO4:func.8">untn</a> <font color="Maroon" title="c3">P</font>; <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="c4">Q</font> is  <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a> </span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1"/>
 not <font color="Maroon" title="c4">Q</font> is  <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</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"> contradiction</span><br/>

<a NAME="E2:39_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a>  <a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T44" onmouseover="rs('ltlaxio1/T44')" onmouseout="rh()">LTLAXIO1:44</a></span>;<br/>
<a NAME="E3:39_1_1"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T18" target="_self" onmouseover="rs('ltlaxio4/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E4:39_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>: 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T52" onmouseover="rs('ltlaxio1/T52')" onmouseout="rh()">LTLAXIO1:52</a></span>;<br/>
<a NAME="E5:39_1_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio1.html#K13" title="LTLAXIO1:func.13">LTL_axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D17" onmouseover="rs('ltlaxio1/D17')" onmouseout="rh()">LTLAXIO1:def 17</a></span>;<br/>
<a NAME="E6:39_1_1"/><span class="kw">then </span>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T42" onmouseover="rs('ltlaxio1/T42')" onmouseout="rh()">LTLAXIO1:42</a></span>;<br/>
<a NAME="E7:39_1_1"/><span class="kw">then </span>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:39_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="ref" href="ltlaxio3.html#D10" onmouseover="rs('ltlaxio3/D10')" onmouseout="rh()">LTLAXIO3:def 10</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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