let p be Instruction-Sequence of SCM+FSA; :: thesis: for J being Program of SCM+FSA
for I being really-closed Program of SCM+FSA
for s being 0 -started State 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 SCM+FSA; :: thesis: for I being really-closed Program of SCM+FSA
for s being 0 -started State 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 SCM+FSA; :: thesis: for s being 0 -started State 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 s be 0 -started State 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 that
A1: I c= p and
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

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;
set sIJ = s;
set pIJ = p +* (I ";" J);
A5: I ";" J c= p +* (I ";" J) by FUNCT_4:25;
let m be Nat; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan (p,s) implies Comput (p,s,m) = Comput ((p +* (I ";" J)),s,m) ) ; :: thesis: S₁[m + 1]
A7: Comput ((p +* (I ";" J)),s,(m + 1)) = Following ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),s,m))) by EXTPRO_1:3;
A8: Comput (p,s,(m + 1)) = Following (p,(Comput (p,s,m))) by EXTPRO_1:3;
A9: p /. (IC (Comput (p,s,m))) = p . (IC (Comput (p,s,m))) by PBOOLE:143;
assume A10: m + 1 <= LifeSpan (p,s) ; :: thesis: Comput (p,s,(m + 1)) = Comput ((p +* (I ";" J)),s,(m + 1))
then A11: IC (Comput (p,s,m)) = IC (Comput ((p +* (I ";" J)),s,m)) by A6, NAT_1:13;
IC s = 0 by MEMSTR_0:def 11;
then IC s in dom I by AFINSQ_1:65;
then A12: IC (Comput (p,s,m)) in dom I by A1, AMISTD_1:21;
A13: CurInstr (p,(Comput (p,s,m))) = I . (IC (Comput (p,s,m))) by A12, A9, A1, GRFUNC_1:2;
A14: (p +* (I ";" J)) /. (IC (Comput ((p +* (I ";" J)),s,m))) = (p +* (I ";" J)) . (IC (Comput ((p +* (I ";" J)),s,m))) by PBOOLE:143;
m < LifeSpan (p,s) by A10, NAT_1:13;
then I . (IC (Comput (p,s,m))) <> halt SCM+FSA by A2, A13, EXTPRO_1:def 15;
then CurInstr (p,(Comput (p,s,m))) = (I ";" J) . (IC (Comput (p,s,m))) by A12, A13, SCMFSA6A:15
.= CurInstr ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),s,m))) by A11, A12, A4, A14, A5, GRFUNC_1:2 ;
hence Comput (p,s,(m + 1)) = Comput ((p +* (I ";" J)),s,(m + 1)) by A6, A10, A8, A7, NAT_1:13; :: thesis: verum

end;

A15: S₁[ 0 ] ;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A15, A3); :: thesis: verum