<?xml version="1.0"?>
<div class="add">

<a NAME="E1:37_2_1"/><i><font color="Green" title="E37">B1</font></i>: 
( <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a> implies  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2_1_1"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>then consider </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">F</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><b> such that </b><br/><a NAME="E3:37_2_1_1"/><i><font color="Green" title="E38">C1</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D13" target="_self" title="MODELC_2:def.13">Def16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>thus </b><a NAME="E4:37_2_1_1"/>
 ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E3:37_2_1_1"><i><font color="Green" title="E38">C1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:37_2_1"/><i><font color="Green" title="E38">B2</font></i>: 
( <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a> implies  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2_1_2"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>then consider </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">F</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><b> such that </b><br/><a NAME="E3:37_2_1_2"/><i><font color="Green" title="E39">C1</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D14" target="_self" title="MODELC_2:def.14">Def17</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>thus </b><a NAME="E4:37_2_1_2"/>
 ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E3:37_2_1_2"><i><font color="Green" title="E39">C1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:37_2_1"/><i><font color="Green" title="E39">B3</font></i>: 
( <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a> implies  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2_1_3"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>then consider </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">F</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><b> such that </b><br/><a NAME="E3:37_2_1_3"/><i><font color="Green" title="E40">C1</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D16" target="_self" title="MODELC_2:def.16">Def19</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>thus </b><a NAME="E4:37_2_1_3"/>
 ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E3:37_2_1_3"><i><font color="Green" title="E40">C1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:37_2_1"/>
( not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a> implies  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2_1_4"/>
( not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>then consider </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">F</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><b> such that </b><br/><a NAME="E3:37_2_1_4"/><i><font color="Green" title="E40">C2</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D17" target="_self" title="MODELC_2:def.17">Def1901</a>, <a class="txt" href="#E1:37"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font></span><br/>

<b>thus </b><a NAME="E4:37_2_1_4"/>
 ex <font color="Olive" title="b1">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E3:37_2_1_4"><i><font color="Green" title="E40">C2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:37_2_1"/>
( ( <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a> implies  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> ) &amp; ( <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a> implies  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> ) &amp; ( <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a> implies  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> ) &amp; ( not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a> implies  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive" title="b2">H1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> ) )
 <b>by </b><i><a class="txt" href="#E1:37_2_1"><i><font color="Green" title="E37">B1</font></i></a>, <a class="txt" href="#E2:37_2_1"><i><font color="Green" title="E38">B2</font></i></a>, <a class="txt" href="#E3:37_2_1"><i><font color="Green" title="E39">B3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
