let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being 0 -started State of SCM+FSA
for I being keeping_0 Program of st not P +* I halts_on s holds
for J being Program of
for k being Element of NAT holds Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
let s be 0 -started State of SCM+FSA; :: thesis: for I being keeping_0 Program of st not P +* I halts_on s holds
for J being Program of
for k being Element of NAT holds Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
let I be keeping_0 Program of ; :: thesis: ( not P +* I halts_on s implies for J being Program of
for k being Element of NAT holds Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k) )
assume A1: not P +* I halts_on s ; :: thesis: for J being Program of
for k being Element of NAT holds Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
let J be Program of ; :: thesis: for k being Element of NAT holds Comput ((P +* I),s,k) = Comput ((P +* (I ";" J)),s,k)
defpred S₁[ Nat] means Comput ((P +* I),s,$1) = Comput ((P +* (I ";" J)),s,$1);
A2: 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 A3: dom I c= dom (I ";" J) by XBOOLE_1:7;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
A4: 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))) ;
A5: 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))) ;
A6: I c= P +* I by FUNCT_4:25;
then A7: IC (Comput ((P +* I),s,m)) in dom I by AMISTD_1:def 10;
assume A8: Comput ((P +* I),s,m) = Comput ((P +* (I ";" J)),s,m) ; :: thesis: S₁[m + 1]
dom (P +* I) = NAT by PARTFUN1:def 2;
then A9: (P +* I) /. (IC (Comput ((P +* I),s,m))) = (P +* I) . (IC (Comput ((P +* I),s,m))) by PARTFUN1:def 6;
dom (P +* (I ";" J)) = NAT by PARTFUN1:def 2;
then A10: (P +* (I ";" J)) /. (IC (Comput ((P +* (I ";" J)),s,m))) = (P +* (I ";" J)) . (IC (Comput ((P +* (I ";" J)),s,m))) by PARTFUN1:def 6;
A11: I ";" J c= P +* (I ";" J) by FUNCT_4:25;
A12: CurInstr ((P +* I),(Comput ((P +* I),s,m))) = I . (IC (Comput ((P +* I),s,m))) by A7, A9, A6, GRFUNC_1:2;
then I . (IC (Comput ((P +* I),s,m))) <> halt SCM+FSA by A1, EXTPRO_1:29;
then CurInstr ((P +* I),(Comput ((P +* I),s,m))) = (I ";" J) . (IC (Comput ((P +* I),s,m))) by A7, A12, SCMFSA6A:15
.= CurInstr ((P +* (I ";" J)),(Comput ((P +* (I ";" J)),s,m))) by A8, A7, A3, A10, A11, GRFUNC_1:2 ;
hence S₁[m + 1] by A8, A4, A5; :: thesis: verum

end;

A13: S₁[ 0 ] ;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A13, A2); :: thesis: verum