let s be 0 -started State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being really-closed Program of st P +* I halts_on s holds
for J being Program of
for k being Nat st k <= LifeSpan ((P +* I),s) holds
Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed Program of st P +* I halts_on s holds
for J being Program of
for k being Nat st k <= LifeSpan ((P +* I),s) holds
Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
let I be really-closed Program of ; :: thesis: ( P +* I halts_on s implies for J being Program of
for k being Nat st k <= LifeSpan ((P +* I),s) holds
Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k) )
assume A1: P +* I halts_on s ; :: thesis: for J being Program of
for k being Nat st k <= LifeSpan ((P +* I),s) holds
Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
let J be Program of ; :: thesis: for k being Nat st k <= LifeSpan ((P +* I),s) holds
Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
A2: I c= P +* I by FUNCT_4:25;
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* I),s) implies Comput ((P +* I),s,$1) = Comput ((P +* (I ";" J)),s,$1) );
A3: for m being Nat st S₁[m] holds
S₁[m + 1]

proof

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

end;

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