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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">H</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>];<br/>
<a NAME="E1:82_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</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">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:82_1_1"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">H</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">H</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E2:82_1_1"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E66">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V8" title="MODELC_2:attr.8">Release</a>  ) implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:82_1_1_1"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
( <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V8" title="MODELC_2:attr.8">Release</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]</span><br/><a NAME="E2:82_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T11" target="_self" onmouseover="rs('modelc_2/T11')" onmouseout="rh()">Th11</a></span>;<br/><a NAME="E3:82_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A9</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a></span>;<br/><a NAME="E4:82_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="modelc_2.html#T11" target="_self" onmouseover="rs('modelc_2/T11')" onmouseout="rh()">Th11</a></span>;<br/><a NAME="E5:82_1_1_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[ <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:82_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E1:82_1_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E3:82_1_1_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V6" title="MODELC_2:attr.6">next</a>  ) implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:82_1_1_2"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
( <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a>  or <font color="Maroon" title="c2">H</font> is  <a href="modelc_2.html#V6" title="MODELC_2:attr.6">next</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]</span><br/><a NAME="E2:82_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T10" target="_self" onmouseover="rs('modelc_2/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E3:82_1_1_2"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[ <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E1:82_1_1_2"><span class="lab"><font color="Green" title="E67">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:82_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E3:82_1_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="modelc_2.html#T2" target="_self" onmouseover="rs('modelc_2/T2')" onmouseout="rh()">Th2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:82_1"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#S4" onmouseover="rs('nat_1/S4')" onmouseout="rh()">NAT_1:sch 4</a>(<a class="txt" href="#E1:82_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">H</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]</span><br/>

<a NAME="E3:82_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:82_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E64">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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