let s be State of SCM+FSA; :: thesis:
set A = NAT ;
let I be InitClosed Program of SCM+FSA; :: thesis:
assume that
A1: ProgramPart (s +* I) halts_on s +* I and
A2: Directed I c= s and
A3: ((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)) c= s ; :: thesis:
set sISA0 = s +* (I +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))));
A4: Initialized I c= s +* (I +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) by A3, Th20;
set IAt = I +* (Start-At (0,SCM+FSA));
dom I misses dom (Start-At (0,SCM+FSA)) by COMPOS_1:140;
then I c= I +* (Start-At (0,SCM+FSA)) by FUNCT_4:33;
then A5: dom I c= dom (I +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
A6: s +* (I +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) = s +* I by A3, Th20;
reconsider sISA0 = s +* (I +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) as State of SCM+FSA ;
set m = LifeSpan ((ProgramPart sISA0),sISA0);
set l1 = IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))));
A7: IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0)))) in dom I by A4, Def1;
set s2 = sISA0 +* (Directed I);

A8: now

set s1 = sISA0 +* (I ';' I);
let k be Element of NAT ; :: thesis:
defpred S₁[ Nat] means ( $1 <= k implies Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),$1), Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),$1) equal_outside NAT );
assume A9: k <= LifeSpan ((ProgramPart sISA0),sISA0) ; :: thesis:
A10: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

A11: Directed I c= I ';' I by SCMFSA6A:55;
let n be Element of NAT ; :: thesis:
A12: dom I c= dom (I ';' I) by SCMFSA6A:56;
assume A13: ( n <= k implies Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n), Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) equal_outside NAT ) ; :: thesis:
T1: ProgramPart (sISA0 +* (I ';' I)) = ProgramPart (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) by AMI_1:123;
T2: ProgramPart (sISA0 +* (Directed I)) = ProgramPart (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) by AMI_1:123;
A14: Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(n + 1)) = Following ((ProgramPart (sISA0 +* (Directed I))),(Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (sISA0 +* (Directed I))),(Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)))),(Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n))) ;
A15: Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),(n + 1)) = Following ((ProgramPart (sISA0 +* (I ';' I))),(Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (sISA0 +* (I ';' I))),(Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)))),(Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) ;
A16: n <= n + 1 by NAT_1:12;
assume A17: n + 1 <= k ; :: thesis:
then A18: IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) = IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) by A13, A16, COMPOS_1:24, XXREAL_0:2;
n <= k by A17, A16, XXREAL_0:2;
then n <= LifeSpan ((ProgramPart sISA0),sISA0) by A9, XXREAL_0:2;
then IC (Comput ((ProgramPart sISA0),sISA0,n)) = IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) by A1, A4, A6, Th18, COMPOS_1:24;
then A19: IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) in dom I by A4, Def1;
then A20: IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) in dom (Directed I) by A18, FUNCT_4:105;
Y: (ProgramPart (sISA0 +* (Directed I))) /. (IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n))) = (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) . (IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n))) by T2, COMPOS_1:38;
Z: (ProgramPart (sISA0 +* (I ';' I))) /. (IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) = (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) . (IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) by T1, COMPOS_1:38;
A21: CurInstr ((ProgramPart (sISA0 +* (Directed I))),(Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n))) = (sISA0 +* (Directed I)) . (IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n))) by Y, AMI_1:54
.= (Directed I) . (IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n))) by A20, FUNCT_4:14 ;
CurInstr ((ProgramPart (sISA0 +* (I ';' I))),(Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) = (sISA0 +* (I ';' I)) . (IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) by Z, AMI_1:54
.= (I ';' I) . (IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) by A12, A19, FUNCT_4:14
.= (Directed I) . (IC (Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n))) by A11, A18, A20, GRFUNC_1:8 ;
hence Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),(n + 1)), Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(n + 1)) equal_outside NAT by A13, A17, A16, A18, A21, A15, A14, SCMFSA6A:32, XXREAL_0:2; :: thesis:

end;

( Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),0) = sISA0 +* (I ';' I) & Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),0) = sISA0 +* (Directed I) ) by EXTPRO_1:3;
then Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),0), Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),0) equal_outside NAT by FUNCT_7:107, FUNCT_7:133;
then A22: S₁[ 0 ] by FUNCT_7:28;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A22, A10);
then A23: Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),k), Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),k) equal_outside NAT ;
Comput ((ProgramPart sISA0),sISA0,k), Comput ((ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),k) equal_outside NAT by A1, A4, A6, A9, Th18;
hence Comput ((ProgramPart sISA0),sISA0,k), Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),k) equal_outside NAT by A23, FUNCT_7:29; :: thesis:

end;

then A24: IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0)))) = IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0)))) by COMPOS_1:24;
TX: ProgramPart sISA0 = ProgramPart (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0)))) by AMI_1:123;
Y: (ProgramPart sISA0) /. (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) = (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0)))) . (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) by TX, COMPOS_1:38;
I +* (Start-At (0,SCM+FSA)) c= Initialized I by Th6;
then I +* (Start-At (0,SCM+FSA)) c= sISA0 by A4, XBOOLE_1:1;
then sISA0 . (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) = (I +* (Start-At (0,SCM+FSA))) . (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) by A7, A5, GRFUNC_1:8;
then A25: I . (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) = sISA0 . (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) by A7, SCMFSA6B:7
.= CurInstr ((ProgramPart sISA0),(Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) by Y, AMI_1:54
.= halt SCM+FSA by A1, A6, EXTPRO_1:def 14 ;
T: ProgramPart (sISA0 +* (Directed I)) = ProgramPart (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0)))) by AMI_1:123;
IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0)))) in dom I by A8, A7, COMPOS_1:24;
then IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0)))) in dom (Directed I) by FUNCT_4:105;
then A26: (sISA0 +* (Directed I)) . (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) = (Directed I) . (IC (Comput ((ProgramPart sISA0),sISA0,(LifeSpan ((ProgramPart sISA0),sISA0))))) by A24, FUNCT_4:14
.= goto (card I) by A7, A25, FUNCT_4:112 ;
Y: (ProgramPart (sISA0 +* (Directed I))) /. (IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0))))) = (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0)))) . (IC (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0))))) by T, COMPOS_1:38;
Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),((LifeSpan ((ProgramPart sISA0),sISA0)) + 1)) = Following ((ProgramPart (sISA0 +* (Directed I))),(Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0))))) by EXTPRO_1:4
.= Exec ((goto (card I)),(Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0))))) by A24, A26, Y, AMI_1:54 ;
then A27: ( ( for a being Int-Location holds (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),((LifeSpan ((ProgramPart sISA0),sISA0)) + 1))) . a = (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0)))) . a ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),((LifeSpan ((ProgramPart sISA0),sISA0)) + 1))) . f = (Comput ((ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan ((ProgramPart sISA0),sISA0)))) . f ) ) by SCMFSA_2:95;
sISA0 +* (Directed I) = s +* (Directed I) by A3, Th20
.= s by A2, FUNCT_4:79 ;
hence DataPart (Comput ((ProgramPart s),s,(LifeSpan ((ProgramPart (s +* I)),(s +* I))))) = DataPart (Comput ((ProgramPart s),s,((LifeSpan ((ProgramPart (s +* I)),(s +* I))) + 1))) by A6, A27, SCMFSA6A:38; :: thesis: