let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for a being read-write Int-Location
for Ig being good Program of SCM+FSA st ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 holds
for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for Ig being good Program of SCM+FSA st ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 holds
for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
let a be read-write Int-Location ; :: thesis: for Ig being good Program of SCM+FSA st ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 holds
for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
let Ig be good Program of SCM+FSA; :: thesis: ( ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) & s . (intloc 0) = 1 implies for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1 )
set I = Ig;
assume that
A1: ( ProperBodyWhile=0 a,Ig,s,p or Ig is parahalting ) and
A2: s . (intloc 0) = 1 ; :: thesis: for k being Element of NAT holds ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1
set SW = StepWhile=0 (a,Ig,p,s);
set PW = p +* (while=0 (a,Ig));
defpred S₁[ Nat] means ((StepWhile=0 (a,Ig,p,s)) . $1) . (intloc 0) = 1;
A4: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: ((StepWhile=0 (a,Ig,p,s)) . k) . (intloc 0) = 1 ; :: thesis: S₁[k + 1]

per cases ( ((StepWhile=0 (a,Ig,p,s)) . k) . a <> 0 or ((StepWhile=0 (a,Ig,p,s)) . k) . a = 0 ) ;

suppose ((StepWhile=0 (a,Ig,p,s)) . k) . a <> 0 ; :: thesis: S₁[k + 1]

then DataPart ((StepWhile=0 (a,Ig,p,s)) . (k + 1)) = DataPart ((StepWhile=0 (a,Ig,p,s)) . k) by Th24;
hence S₁[k + 1] by A5, SCMFSA6A:38; :: thesis: verum

end;

suppose A6: ((StepWhile=0 (a,Ig,p,s)) . k) . a = 0 ; :: thesis: S₁[k + 1]

set Ins = NAT ;
set IS = Initialize Ig;
set SWkIS = Initialize ((StepWhile=0 (a,Ig,p,s)) . k);
set PWIS = (p +* (while=0 (a,Ig))) +* Ig;
A7: dom (((StepWhile=0 (a,Ig,p,s)) . k) | NAT) c= NAT by RELAT_1:87;
NAT misses Int-Locations \/ FinSeq-Locations by SCMFSA_2:13, SCMFSA_2:14, XBOOLE_1:70;
then A8: dom (((StepWhile=0 (a,Ig,p,s)) . k) | NAT) misses Int-Locations \/ FinSeq-Locations by A7, XBOOLE_1:63;
set SWkI = ((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1));
set PWI = (p +* (while=0 (a,Ig))) +* Ig;
set ISWk = Initialized ((StepWhile=0 (a,Ig,p,s)) . k);
A9: DataPart ((StepWhile=0 (a,Ig,p,s)) . k) = DataPart (Initialized ((StepWhile=0 (a,Ig,p,s)) . k)) by A5, SCMFSA8C:27;
A10: ProperBodyWhile=0 a,Ig,s,p by A1, Th19;
then A11: Ig is_closed_on (StepWhile=0 (a,Ig,p,s)) . k,p +* (while=0 (a,Ig)) by A6, Def1;
Ig is_halting_on (StepWhile=0 (a,Ig,p,s)) . k,p +* (while=0 (a,Ig)) by A6, A10, Def1;
then A13: Ig is_halting_on Initialized ((StepWhile=0 (a,Ig,p,s)) . k),p +* (while=0 (a,Ig)) by A11, A9, SCMFSA8B:8;
XX: Start-At (0,SCM+FSA) c= Initialized ((StepWhile=0 (a,Ig,p,s)) . k) by SCMFSA6B:4;
((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) = ((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1))
.= Initialized ((StepWhile=0 (a,Ig,p,s)) . k)
.= Initialize (Initialized ((StepWhile=0 (a,Ig,p,s)) . k)) by XX, FUNCT_4:104 ;
then B14: (p +* (while=0 (a,Ig))) +* Ig halts_on ((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) by A13, SCMFSA7B:def 8;
A15: (p +* (while=0 (a,Ig))) +* Ig halts_on ((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) by B14;
A16: ((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1)) = Initialize ((StepWhile=0 (a,Ig,p,s)) . k) by A5, SCMFSA8C:18;
A17: DataPart (IExec (Ig,(p +* (while=0 (a,Ig))),((StepWhile=0 (a,Ig,p,s)) . k))) = DataPart ((Result (((p +* (while=0 (a,Ig))) +* Ig),(((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1))))) +* (((StepWhile=0 (a,Ig,p,s)) . k) | NAT)) by SCMFSA6B:def 1
.= DataPart (Result (((p +* (while=0 (a,Ig))) +* Ig),(((StepWhile=0 (a,Ig,p,s)) . k) +* (Initialize ((intloc 0) .--> 1))))) by A8, FUNCT_4:76, SCMFSA_2:127
.= DataPart (Comput (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)),(LifeSpan (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)))))) by A16, A15, EXTPRO_1:23 ;
Ig is_closed_on Initialized ((StepWhile=0 (a,Ig,p,s)) . k),p +* (while=0 (a,Ig)) by A11, A9, SCMFSA8B:6;
then DataPart ((StepWhile=0 (a,Ig,p,s)) . (k + 1)) = DataPart (IExec (Ig,(p +* (while=0 (a,Ig))),((StepWhile=0 (a,Ig,p,s)) . k))) by A5, A6, A13, Th25;
hence ((StepWhile=0 (a,Ig,p,s)) . (k + 1)) . (intloc 0) = (Comput (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)),(LifeSpan (((p +* (while=0 (a,Ig))) +* Ig),(Initialize ((StepWhile=0 (a,Ig,p,s)) . k)))))) . (intloc 0) by A17, SCMFSA6A:38
.= 1 by A5, A11, SCMFSA8C:97 ;
:: thesis: verum

end;

end;

end;

A18: S₁[ 0 ] by A2, SCMFSA_9:def 4;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A18, A4); :: thesis: verum