let s be 0 -started State of SCM+FSA; :: thesis: for I being really-closed Program of
for J being Program of
for P being Instruction-Sequence of SCM+FSA st I c= P & P halts_on s holds
for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ";" J)),s,m)
let I be really-closed Program of ; :: thesis: for J being Program of
for P being Instruction-Sequence of SCM+FSA st I c= P & P halts_on s holds
for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ";" J)),s,m)
let J be Program of ; :: thesis: for P being Instruction-Sequence of SCM+FSA st I c= P & P halts_on s holds
for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ";" J)),s,m)
let P be Instruction-Sequence of SCM+FSA; :: thesis: ( I c= P & P halts_on s implies for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ";" J)),s,m) )
assume A1: I c= P ; :: thesis: ( not P halts_on s or for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ";" J)),s,m) )
assume A2: P halts_on s ; :: thesis: for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ";" J)),s,m)
defpred S₁[ Nat] means ( $1 <= LifeSpan (P,s) implies Comput (P,s,$1) = Comput ((P +* (I ";" J)),s,$1) );
A3: for m being Nat st S₁[m] holds
S₁[m + 1]

proof

let m be Nat; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A4: ( m <= LifeSpan (P,s) implies Comput (P,s,m) = Comput ((P +* (I ";" J)),s,m) ) ; :: thesis: S₁[m + 1]
A5: dom (I ";" J) = (dom I) \/ (dom (Reloc (J,(card I)))) by SCMFSA6A:39;
A6: ( {} c= Comput ((P +* (I ";" J)),s,m) & dom I c= dom (I ";" J) ) by A5, XBOOLE_1:2, XBOOLE_1:7;
A7: Comput (P,s,(m + 1)) = Following (P,(Comput (P,s,m))) by EXTPRO_1:3;
A8: Comput ((P +* (I ";" J)),s,(m + 1)) = Following ((P +* (I ";" J)),(Comput ((P +* (I ";" J)),s,m))) by EXTPRO_1:3;
IC s = 0 by MEMSTR_0:def 11;
then IC s in dom I by AFINSQ_1:65;
then A9: IC (Comput (P,s,m)) in dom I by A1, AMISTD_1:21;
dom P = NAT by PARTFUN1:def 2;
then A10: CurInstr (P,(Comput (P,s,m))) = P . (IC (Comput (P,s,m))) by PARTFUN1:def 6
.= I . (IC (Comput (P,s,m))) by A9, A1, GRFUNC_1:2 ;
assume A11: m + 1 <= LifeSpan (P,s) ; :: thesis: Comput (P,s,(m + 1)) = Comput ((P +* (I ";" J)),s,(m + 1))
A12: I ";" J c= P +* (I ";" J) by FUNCT_4:25;
A13: dom (P +* (I ";" J)) = NAT by PARTFUN1:def 2;
m < LifeSpan (P,s) by A11, NAT_1:13;
then I . (IC (Comput (P,s,m))) <> halt SCM+FSA by A2, A10, EXTPRO_1:def 15;
then CurInstr (P,(Comput (P,s,m))) = (I ";" J) . (IC (Comput (P,s,m))) by A9, A10, SCMFSA6A:15
.= (P +* (I ";" J)) . (IC (Comput (P,s,m))) by A9, A6, A12, GRFUNC_1:2
.= CurInstr ((P +* (I ";" J)),(Comput ((P +* (I ";" J)),s,m))) by A13, A11, A4, NAT_1:13, PARTFUN1:def 6 ;
hence Comput (P,s,(m + 1)) = Comput ((P +* (I ";" J)),s,(m + 1)) by A7, A8, A4, A11, NAT_1:13; :: thesis: verum

end;

A14: S₁[ 0 ] ;
thus for m being Nat holds S₁[m] from NAT_1:sch 2(A14, A3); :: thesis: verum