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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</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="c3">H</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> iff ( <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> ) )</span><br/>





<span class="kw">thus </span><a NAME="E1:47"/>
(  not <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> or <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</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="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> ) implies <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">Z</font> = <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font>;<br/>
<div><span class="lab"><font color="Green" title="E42">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1"><span class="kw">now </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"> (  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <font color="Maroon" title="c3">H</font> &amp;  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <font color="Maroon" title="c3">H</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:47_1_1"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
( <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <font color="Maroon" title="c3">H</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"> contradiction</span><br/><a NAME="E2:47_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T12" target="_self" onmouseover="rs('modelc_2/T12')" onmouseout="rh()">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="modelc_2.html#T12" target="_self" onmouseover="rs('modelc_2/T12')" onmouseout="rh()">Th12</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_2"><span class="kw">now </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">  for <font color="Olive" title="b1">H1</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> holds <br/> (  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Olive" title="b1">H1</font> &amp;  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Maroon" title="c3">H</font> &amp;  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Olive" title="b1">H1</font> &amp;  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">H</font> &amp;  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Olive" title="b1">H1</font> &amp;  not <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Maroon" title="c3">H</font> )</span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c5">H1</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><span class="kw"> such that </span><a NAME="E2:47_1_2"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
( <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Maroon" title="c3">H</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"> contradiction</span><br/><a NAME="E3:47_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T12" target="_self" onmouseover="rs('modelc_2/T12')" onmouseout="rh()">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="modelc_2.html#T12" target="_self" onmouseover="rs('modelc_2/T12')" onmouseout="rh()">Th12</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/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:47_1"/>
<font color="Maroon" title="c3">H</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</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="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> )</span><br/>

<a NAME="E4:47_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <font color="Maroon" title="c3">H</font> or  ex <font color="Olive" title="b1">H1</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <br/>( <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Olive" title="b1">H1</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Olive" title="b1">H1</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Olive" title="b1">H1</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Olive" title="b1">H1</font> or <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Maroon" title="c3">H</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:47_1"/>
( <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="modelc_2.html#E32"><span class="lab"><font color="Green" title="E18">Lm12</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:47"/>
( ( <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> ) implies <font color="Maroon" title="c3">H</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c2">G</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"> verum</span><br/>


</div>
