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

<span class="kw">let </span><font color="Maroon" title="c1">I</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L1" title="FSM_1:struct.1">FSM</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b3">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Olive" title="b2">S</font>  st <font color="Olive" title="b2">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  &amp; <font color="Olive" title="b3">q</font> <a href="fsm_2.html#R1" title="FSM_2:pred.1">is_accessible_via</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b4">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b5">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b5">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> holds <br/>( <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b5">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m</font> &amp; ( for <font color="Olive" title="b6">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b5">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L1" title="FSM_1:struct.1">FSM</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b1">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  &amp; <font color="Olive" title="b2">q</font> <a href="fsm_2.html#R1" title="FSM_2:pred.1">is_accessible_via</a> <font color="Maroon" title="c2">s</font> holds <br/> ex <font color="Olive" title="b3">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b4">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b4">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> holds <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b4">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> &amp; ( for <font color="Olive" title="b5">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b4">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L1" title="FSM_1:struct.1">FSM</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">S</font>  st <font color="Maroon" title="c3">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  &amp; <font color="Olive" title="b1">q</font> <a href="fsm_2.html#R1" title="FSM_2:pred.1">is_accessible_via</a> <font color="Maroon" title="c2">s</font> holds <br/> ex <font color="Olive" title="b2">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b3">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> holds <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp; ( for <font color="Olive" title="b4">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">q</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">q</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="fsm_2.html#R1" title="FSM_2:pred.1">is_accessible_via</a> <font color="Maroon" title="c2">s</font> implies  ex <font color="Olive" title="b1">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> holds <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> &amp; ( for <font color="Olive" title="b3">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c3">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">q</font> <a href="fsm_2.html#R1" title="FSM_2:pred.1">is_accessible_via</a> <font color="Maroon" title="c2">s</font> or  ex <font color="Olive" title="b1">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> holds <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> &amp; ( for <font color="Olive" title="b3">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> ) ) )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><a NAME="E3:18"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c4">q</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="fsm_2.html#D2" target="_self" title="FSM_2:def.2">FSM_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> holds <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> &amp; ( for <font color="Olive" title="b3">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> ) )</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>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 );<br/>
<a NAME="E4:18"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="E5:18"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="fsm_1.html#D3" title="FSM_1:def.3">FSM_1:def 3</a></span>;<br/>
<a NAME="E6:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 ex <font color="Olive" title="b1">m</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">m</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:18"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>]
 <span class="kw">and </span><br/><a NAME="E9:18"/><span class="lab"><font color="Green" title="E16">A6</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" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A4</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">m</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">m</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> holds <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> &amp; ( for <font color="Olive" title="b2">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">w1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c8">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> implies ( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> &amp; ( for <font color="Olive" title="b1">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:18"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font>
 <span class="kw">and </span><br/><a NAME="E12:18"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c8">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> &amp; ( for <font color="Olive" title="b1">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> ) )</span><br/>

<a NAME="E13:18"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>;<br/>
<a NAME="E14:18"/><span class="kw">then </span>
 <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="fsm_2.html#T4" target="_self" title="FSM_2:th.4">Th4</a></span>;<br/>
<a NAME="E15:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
(  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font> or  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E20">A10</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E21">A11</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>
;<br/>
<a NAME="E18:18"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E16:18"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E19:18"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:18"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E18:18"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> implies <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E21:18"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font></span><br/>

<a NAME="E22:18"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E23:18"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E21:18"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:18"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E21:18"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E27">A17</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>
;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>
;<br/>
<a NAME="E27:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:18"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E23:18"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E28:18"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:18"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E24:18"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E25:18"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E29:18"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E30:18"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E27:18"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E28:18"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:18"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E21:18"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E22:18"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E23:18"><span class="lab"><font color="Green" title="E25">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
