let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: for s1, s2 being State of SCMPDS
for I being Program of SCMPDS st I is_closed_on s1,P1 & DataPart s1 = DataPart s2 holds
for k being Element of NAT holds
( Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) )
let s1, s2 be State of SCMPDS; :: thesis: for I being Program of SCMPDS st I is_closed_on s1,P1 & DataPart s1 = DataPart s2 holds
for k being Element of NAT holds
( Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) )
let I be Program of SCMPDS; :: thesis: ( I is_closed_on s1,P1 & DataPart s1 = DataPart s2 implies for k being Element of NAT holds
( Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) ) )
assume A1: I is_closed_on s1,P1 ; :: thesis: ( not DataPart s1 = DataPart s2 or for k being Element of NAT holds
( Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) ) )
set pI = stop I;
set ss1 = Initialize s1;
set PP1 = P1 +* (stop I);
set ss2 = Initialize s2;
set PP2 = P2 +* (stop I);
A5: stop I c= P2 +* (stop I) by FUNCT_4:25;
A6: stop I c= P1 +* (stop I) by FUNCT_4:25;
assume A7: DataPart s1 = DataPart s2 ; :: thesis: for k being Element of NAT holds
( Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) )
let k be Element of NAT ; :: thesis: ( Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) )
A8: IC (Comput ((P1 +* (stop I)),(Initialize s1),k)) in dom (stop I) by A1, SCMPDS_6:def 2;
A9: I is_closed_on s2,P2 by A1, A7, SCMPDS_6:22;
then A10: for m being Element of NAT st m < k holds
IC (Comput ((P2 +* (stop I)),(Initialize s2),m)) in dom (stop I) by SCMPDS_6:def 2;
Initialize s1 = Initialize s2 by A7, MEMSTR_0:80;
hence K: Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) by A6, A5, SCMPDS_4:21, A10; :: thesis: CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k)))
A12: IC (Comput ((P2 +* (stop I)),(Initialize s2),k)) in dom (stop I) by A9, SCMPDS_6:def 2;
thus CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) = (P2 +* (stop I)) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),k))) by PBOOLE:143
.= (stop I) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),k))) by A5, A12, GRFUNC_1:2
.= (P1 +* (stop I)) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),k))) by A6, K, A8, GRFUNC_1:2
.= CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) by PBOOLE:143 ; :: thesis: verum