let s be 0 -started State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being paraclosed Program of st P +* I halts_on s holds
for J being Program of
for k being Element of 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 paraclosed Program of st P +* I halts_on s holds
for J being Program of
for k being Element of NAT st k <= LifeSpan ((P +* I),s) holds
Comput ((P +* I),s,k) = Comput ((P +* (I ';' J)),s,k)
let I be paraclosed Program of ; :: thesis: ( P +* I halts_on s implies for J being Program of
for k being Element of 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 Element of 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 Element of NAT st k <= LifeSpan ((P +* I),s) holds
Comput ((P +* I),s,k) = Comput ((P +* (I ';' J)),s,k)
A3: 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) );
A5: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom (Directed I)) \/ (dom (Reloc (J,(card I)))) by FUNCT_4:def 1
.= (dom I) \/ (dom (Reloc (J,(card I)))) by FUNCT_4:99 ;
then A6: dom I c= dom (I ';' J) by XBOOLE_1:7;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A7: ( m <= LifeSpan ((P +* I),s) implies Comput ((P +* I),s,m) = Comput ((P +* (I ';' J)),s,m) ) ; :: thesis: S₁[m + 1]
A8: 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))) ;
A9: 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))) ;
A10: IC (Comput ((P +* I),s,m)) in dom I by A3, AMISTD_1:def 10;
A11: I c= P +* I by FUNCT_4:25;
dom (P +* I) = NAT by PARTFUN1:def 2;
then A12: 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 A10, A11, GRFUNC_1:2 ;
assume A13: m + 1 <= LifeSpan ((P +* I),s) ; :: thesis: Comput ((P +* I),s,(m + 1)) = Comput ((P +* (I ';' J)),s,(m + 1))
A15: I ';' J c= P +* (I ';' J) by FUNCT_4:25;
A16: dom (P +* (I ';' J)) = NAT by PARTFUN1:def 2;
m < LifeSpan ((P +* I),s) by A13, NAT_1:13;
then I . (IC (Comput ((P +* I),s,m))) <> halt SCM+FSA by A1, A12, EXTPRO_1:def 15;
then CurInstr ((P +* I),(Comput ((P +* I),s,m))) = (I ';' J) . (IC (Comput ((P +* I),s,m))) by A10, A12, SCMFSA6A:15
.= (P +* (I ';' J)) . (IC (Comput ((P +* (I ';' J)),s,m))) by A13, A10, A6, A15, A7, GRFUNC_1:2, NAT_1:13
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m))) by A16, PARTFUN1:def 6 ;
hence Comput ((P +* I),s,(m + 1)) = Comput ((P +* (I ';' J)),s,(m + 1)) by A8, A9, A7, A13, NAT_1:13; :: thesis: verum

end;

A18: ( Comput ((P +* I),s,0) = s & Comput ((P +* (I ';' J)),s,0) = s ) by EXTPRO_1:2;
A21: S₁[ 0 ] by A18;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A21, A5); :: thesis: verum