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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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 <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Olive" title="b3">v</font></span>)</span>  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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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 <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c4">v</font></span>)</span>  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/><span class="kw">let </span><font color="Maroon" title="c5">q</font> be   <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c4">v</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>









<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>  <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/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:115"/><span class="lab"><font color="Green" title="E100">A1</font></span>: 
<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)
 <span class="kw">and </span><br/><a NAME="E2:115"/><span class="lab"><font color="Green" title="E101">A2</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( (  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="E3:115"/><span class="lab"><font color="Green" title="E102">A3</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:115_1"/><span class="lab"><font color="Green" title="E103">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:115_1_1"/><span class="lab"><font color="Green" title="E104">A5</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E105">A6</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E3:115_1_1"/><span class="lab"><font color="Green" title="E106">A7</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:115_1_1_1"/><span class="lab"><font color="Green" title="E107">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E2:115_1_1_1"/><span class="lab"><font color="Green" title="E108">A9</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:115_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 href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1"><span class="lab"><font color="Green" title="E105">A6</font></span></a>, <a class="txt" href="#E2:115_1_1_1"><span class="lab"><font color="Green" title="E108">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:115_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1"><span class="lab"><font color="Green" title="E104">A5</font></span></a>, <a class="txt" href="#E1:115_1_1_1"><span class="lab"><font color="Green" title="E107">A8</font></span></a></span>; <a class="txt" 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>
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:115_1_1_2"/><span class="lab"><font color="Green" title="E107">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E2:115_1_1_2"/><span class="lab"><font color="Green" title="E108">A11</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:115_1_1_2"/><span class="lab"><font color="Green" title="E109">A12</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1_2"><span class="lab"><font color="Green" title="E108">A11</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1_1_2"><span class="lab"><font color="Green" title="E109">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_1"><span class="kw">suppose </span></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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">hence </span><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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E103">A4</font></span></a>, <a class="txt" href="#E3:115_1_1"><span class="lab"><font color="Green" title="E106">A7</font></span></a>, <a class="txt" href="#E1:115_1_1_2"><span class="lab"><font color="Green" title="E107">A10</font></span></a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E110">A13</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1_2"><span class="lab"><font color="Green" title="E107">A10</font></span></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E110">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E111">A14</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><a NAME="E2:115_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E112">A15</font></span>: 
( <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) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E101">A2</font></span></a></span>;<br/><a NAME="E3:115_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E113">A16</font></span>: 
(  ex <font color="Olive" title="b1">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> st <font color="Olive" title="b1">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> ) &amp;  ex <font color="Olive" title="b1">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> st <font color="Olive" title="b1">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(2) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">Th52</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">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><span class="kw"> such that </span><br/><a NAME="E5:115_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E114">A17</font></span>: 
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(1)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">Th52</a></span>;<br/><a NAME="E6:115_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="c9">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1"><span class="lab"><font color="Green" title="E104">A5</font></span></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E111">A14</font></span></a>, <a class="txt" href="#E5:115_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E114">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:115_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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E100">A1</font></span></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E110">A13</font></span></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E111">A14</font></span></a>, <a class="txt" href="#E5:115_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E114">A17</font></span></a>, <a class="txt" href="#E3:115_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E113">A16</font></span></a>, <a class="txt" href="#E2:115_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E112">A15</font></span></a>, <a class="txt" href="modelc_3.html#E100"><span class="lab"><font color="Green" title="E85">Lm30</font></span></a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E111">A18</font></span>: 
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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">set </span><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/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><span class="kw">set </span><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/><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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E111">A18</font></span></a></span>;<br/><a NAME="E4:115_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">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><span class="kw"> such that </span><br/><a NAME="E7:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E113">A20</font></span>: 
<font color="Maroon" title="c13">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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">Th52</a></span>;<br/><a NAME="E8:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E114">A21</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<br/><a NAME="E9:115_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1"><span class="lab"><font color="Green" title="E105">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:115_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A22</font></span>: 
 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="c13">sm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1"><span class="lab"><font color="Green" title="E104">A5</font></span></a>, <a class="txt" href="#E4:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="txt" href="#E7:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a></span>;<br/><a NAME="E11:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E116">A23</font></span>: 
(  ex <font color="Olive" title="b1">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> st <font color="Olive" title="b1">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) &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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E101">A2</font></span></a>, <a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">Th52</a></span>;<br/><a NAME="E12:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E117">A24</font></span>: 
(  ex <font color="Olive" title="b1">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> st <font color="Olive" title="b1">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>) &amp; <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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E101">A2</font></span></a>, <a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">Th52</a></span>;<br/><a NAME="E13:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E118">A25</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E101">A2</font></span></a></span>;<br/><a NAME="E14:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E119">A26</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E101">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">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><span class="kw"> such that </span><br/><a NAME="E16:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E120">A27</font></span>: 
<font color="Maroon" title="c14">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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T52" target="_self" title="MODELC_3:th.52">Th52</a></span>;<br/><a NAME="E17:115_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E121">A28</font></span>: 
 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">sk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1"><span class="lab"><font color="Green" title="E104">A5</font></span></a>, <a class="txt" href="#E1:115_1_1_2"><span class="lab"><font color="Green" title="E107">A10</font></span></a>, <a class="txt" href="#E2:115_1_1_2"><span class="lab"><font color="Green" title="E108">A11</font></span></a>, <a class="txt" href="#E1:115_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E110">A13</font></span></a>, <a class="txt" href="#E16:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A27</font></span></a></span>;<br/><a NAME="E18:115_1_1_2_1_1_2_1_1_2"/>
<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="c13">sm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E103">A4</font></span></a>, <a class="txt" href="#E3:115_1_1"><span class="lab"><font color="Green" title="E106">A7</font></span></a>, <a class="txt" href="#E4:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="txt" href="#E7:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="txt" href="#E8:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A21</font></span></a></span>;<br/><a NAME="E19:115_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<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">sk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="txt" href="#E16:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A27</font></span></a>, <a class="txt" href="#E10:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E115">A22</font></span></a>, <a class="txt" href="#E12:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A24</font></span></a>, <a class="txt" href="#E13:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A25</font></span></a>, <a class="txt" href="modelc_3.html#E100"><span class="lab"><font color="Green" title="E85">Lm30</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E20:115_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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E110">A13</font></span></a>, <a class="txt" href="#E7:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="txt" href="#E16:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A27</font></span></a>, <a class="txt" href="#E17:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A28</font></span></a>, <a class="txt" href="#E11:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A23</font></span></a>, <a class="txt" href="#E14:115_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A26</font></span></a>, <a class="txt" href="modelc_3.html#E100"><span class="lab"><font color="Green" title="E85">Lm30</font></span></a></span>; <a class="txt" 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="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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:()"><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="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:115_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:()"><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">hence </span><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:()"><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">hence </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:115"/><span class="lab"><font color="Green" title="E103">A29</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E5:115"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E3:115"><span class="lab"><font color="Green" title="E102">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:115"/>
(  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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
