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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">v</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><br/> for <font color="Olive" title="b4">q</font> being  <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Olive" title="b3">v</font>  st <font color="Olive" title="b1">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Olive" title="b2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>,<font color="Olive" title="b3">v</font></span>)</span> &amp; ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b3">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span>,<font color="Olive" title="b3">v</font> ) &amp; ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> not <font color="Olive" title="b2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span>,<font color="Olive" title="b3">v</font></span>)</span> ) holds <br/>for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span>,<font color="Olive" title="b3">v</font></span>)</span> &amp; <font color="Olive" title="b1">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Olive" title="b2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span>,<font color="Olive" title="b3">v</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font>, <font color="Maroon" title="c4">v</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q</font> being  <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c4">v</font>  st <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<font color="Maroon" title="c4">v</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) holds <br/>for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c5">q</font> be   <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c4">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )</span><br/>









<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b> <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c4">v</font>;<br/>
<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E100">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(1) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 holds <br/> not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) );<br/>
<a NAME="E2:115"/><i><font color="Green" title="E101">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:115"/><i><font color="Green" title="E102">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:115_1"/><i><font color="Green" title="E103">B1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>set </b><font color="Maroon" title="c7">k1</font> = <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E2:115_1"/>
( ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:115_1_1"/><i><font color="Green" title="E104">C1</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) )</span><br/>

<a NAME="E2:115_1_1"/><i><font color="Green" title="E105">C2</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:115_1_1"/><i><font color="Green" title="E106">C3</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/> not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> implies  not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )</span><br/>

<b>assume </b><a NAME="E1:115_1_1_1"/><i><font color="Green" title="E107">D1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>)</span><br/>

<a NAME="E2:115_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:115_1_1_1"><i><font color="Green" title="E107">D1</font></i></a>, <a class="txt" href="#E2:115_1_1"><i><font color="Green" title="E105">C2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:115_1_1_1"/>
not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>)
 <b>by </b><i><a class="txt" href="#E1:115_1_1"><i><font color="Green" title="E104">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="E4:115_1_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) ) )</span><br/>

<b>assume </b><a NAME="E1:115_1_1_2"/><i><font color="Green" title="E107">D1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )</span><br/>

<a NAME="E2:115_1_1_2"/><i><font color="Green" title="E108">D101</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="#E1:115_1_1_2"><i><font color="Green" title="E107">D1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_1_1_2_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> )
 </span><b>by </b><i><a class="txt" href="#E2:115_1_1_2"><i><font color="Green" title="E108">D101</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:115_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )</span><br/><div class="add"><b>hence </b><a NAME="E2:115_1_1_2_1_1_1"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )
 <b>by </b><i><a class="txt" href="#E1:115_1_1_2"><i><font color="Green" title="E107">D1</font></i></a>, <a class="txt" href="#E3:115_1_1"><i><font color="Green" title="E106">C3</font></i></a>, <a class="txt" href="#E1:115_1"><i><font color="Green" title="E103">B1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:115_1_1_2_1_1_2"/><i><font color="Green" title="E109">D2</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> )
 </span><b>by </b><i><a class="txt" href="#E1:115_1_1_2"><i><font color="Green" title="E107">D1</font></i></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2"><i><font color="Green" title="E109">D2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:115_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E110">E1</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c9">s0</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a> <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font><b> such that </b><br/><a NAME="E3:115_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E111">E100</font></i>: 
<font color="Maroon" title="c9">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
 <b>by </b><i><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">ThLTLStates04</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">s1</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a> <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font><b> such that </b><br/><a NAME="E5:115_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E112">E101</font></i>: 
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(1)
 <b>by </b><i><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">ThLTLStates04</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">s2</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a> <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font><b> such that </b><br/><a NAME="E7:115_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E113">E102</font></i>: 
<font color="Maroon" title="c11">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(2)
 <b>by </b><i><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">ThLTLStates04</a></i>;<br/><a NAME="E8:115_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E114">E104</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) &amp; <font color="Maroon">H<sub>1</sub></font>(1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>(1) )
 <b>by </b><i><a class="txt" href="#E1:115"><i><font color="Green" title="E100">A1</font></i></a></i>;<br/><a NAME="E9:115_1_1_2_1_1_2_1_1_1"/>
not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c10">s1</font>
 <b>by </b><i><a class="txt" href="#E1:115_1_1"><i><font color="Green" title="E104">C1</font></i></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E110">E1</font></i></a>, <a class="txt" href="#E5:115_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E112">E101</font></i></a></i>;<br/><b>hence </b><a NAME="E10:115_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )
 <b>by </b><i><a class="txt" href="modelc_3.html#E100"><i><font color="Green" title="E85">lemUntil01</font></i></a>, <a class="txt" href="#E1:115"><i><font color="Green" title="E100">A1</font></i></a>, <a class="txt" href="#E8:115_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E114">E104</font></i></a>, <a class="txt" href="#E3:115_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E111">E100</font></i></a>, <a class="txt" href="#E5:115_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E112">E101</font></i></a>, <a class="txt" href="#E7:115_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E113">E102</font></i></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2"><i><font color="Green" title="E109">D2</font></i></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E110">E1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:115_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E110">E2</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )</span><br/><div class="add"><b>set </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c6">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><b>reconsider </b><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c6">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E110">E2</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/><a NAME="E3:115_1_1_2_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E110">E2</font></i></a></i>;<br/><a NAME="E4:115_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E111">E201</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><b>reconsider </b><font color="Maroon" title="c12">m1</font> = <font color="Maroon" title="c10">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E4:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E111">E201</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">sm1</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a> <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font><b> such that </b><br/><a NAME="E7:115_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E112">Esm1</font></i>: 
<font color="Maroon" title="c13">sm1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">m1</font>)
 <b>by </b><i><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">ThLTLStates04</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">sm</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a> <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font><b> such that </b><br/><a NAME="E9:115_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E113">Esm</font></i>: 
<font color="Maroon" title="c14">sm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">m</font>)
 <b>by </b><i><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">ThLTLStates04</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">sk</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a> <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font><b> such that </b><br/><a NAME="E11:115_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E114">Esk</font></i>: 
<font color="Maroon" title="c15">sk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">k</font>)
 <b>by </b><i><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">ThLTLStates04</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">sk1</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a> <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c4">v</font><b> such that </b><br/><a NAME="E13:115_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E115">Esk1</font></i>: 
<font color="Maroon" title="c16">sk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <b>by </b><i><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">ThLTLStates04</a></i>;<br/><a NAME="E14:115_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E116">E202</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></i>;<br/><a NAME="E15:115_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E117">E203</font></i>: 
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">sm</font> &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">sm</font> )
 <b>by </b><i><a class="txt" href="#E4:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E111">E201</font></i></a>, <a class="txt" href="#E3:115_1_1"><i><font color="Green" title="E106">C3</font></i></a>, <a class="txt" href="#E1:115_1"><i><font color="Green" title="E103">B1</font></i></a>, <a class="txt" href="#E9:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E113">Esm</font></i></a></i>;<br/><a NAME="E16:115_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="#E14:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E116">E202</font></i></a>, <a class="txt" href="#E2:115_1_1"><i><font color="Green" title="E105">C2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:115_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E118">E204</font></i>: 
not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">sm</font>
 <b>by </b><i><a class="txt" href="#E4:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E111">E201</font></i></a>, <a class="txt" href="#E1:115_1_1"><i><font color="Green" title="E104">C1</font></i></a>, <a class="txt" href="#E9:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E113">Esm</font></i></a></i>;<br/><a NAME="E18:115_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">m1</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">m</font>) )
 <b>by </b><i><a class="txt" href="#E1:115"><i><font color="Green" title="E100">A1</font></i></a></i>;<br/><a NAME="E19:115_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E119">E205</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c15">sk</font>
 <b>by </b><i><a class="txt" href="#E7:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E112">Esm1</font></i></a>, <a class="txt" href="#E9:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E113">Esm</font></i></a>, <a class="txt" href="#E11:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E114">Esk</font></i></a>, <a class="txt" href="modelc_3.html#E100"><i><font color="Green" title="E85">lemUntil01</font></i></a>, <a class="txt" href="#E15:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E117">E203</font></i></a>, <a class="txt" href="#E17:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E118">E204</font></i></a></i>;<br/><a NAME="E20:115_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E120">E207</font></i>: 
not <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c15">sk</font>
 <b>by </b><i><a class="txt" href="#E1:115_1_1_2"><i><font color="Green" title="E107">D1</font></i></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2"><i><font color="Green" title="E109">D2</font></i></a>, <a class="txt" href="#E1:115_1_1"><i><font color="Green" title="E104">C1</font></i></a>, <a class="txt" href="#E11:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E114">Esk</font></i></a></i>;<br/><a NAME="E21:115_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">m</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">k</font>) )
 <b>by </b><i><a class="txt" href="#E1:115"><i><font color="Green" title="E100">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E22:115_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )
 <b>by </b><i><a class="txt" href="#E9:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E113">Esm</font></i></a>, <a class="txt" href="#E11:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E114">Esk</font></i></a>, <a class="txt" href="#E13:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E115">Esk1</font></i></a>, <a class="txt" href="modelc_3.html#E100"><i><font color="Green" title="E85">lemUntil01</font></i></a>, <a class="txt" href="#E19:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E119">E205</font></i></a>, <a class="txt" href="#E20:115_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E120">E207</font></i></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2"><i><font color="Green" title="E109">D2</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:115_1_1_2_1_1_2"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )
 ; <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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:115_1_1_2"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">i</font>) )
 ; <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="E5:115_1_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) )
 ; <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="E3:115_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E4:115"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:115"><i><font color="Green" title="E101">A2</font></i></a>, <a class="txt" href="#E3:115"><i><font color="Green" title="E102">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:115"/>
(  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Maroon" title="c2">F</font> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
