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<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_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">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  st <font color="Olive" title="b1">X</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c3">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Olive" title="b1">X</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</font> holds <br/><font color="Olive" title="b1">X</font> <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="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_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="c7">X</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c3">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c7">X</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</font> implies <font color="Maroon" title="c7">X</font> <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="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></span>)</span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c3">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">and </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</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="c7">X</font> <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="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></span>)</span></span><br/>

<a NAME="E3:61"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T45" target="_self" onmouseover="rs('ltlaxio2/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<a NAME="E4:61"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></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="E5:61"/><span class="kw">then </span>
<font color="Maroon" title="c7">X</font> <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="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></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="E6:61"/><span class="kw">then </span>
<font color="Maroon" title="c7">X</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:61"/>
<font color="Maroon" title="c7">X</font> <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="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon" title="c6">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E39">A2</font></span></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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