let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for s1, s2 being State of SCM+FSA
for I being really-closed Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_halting_on s1,P1 holds
I is_halting_on s2,P2
let s1, s2 be State of SCM+FSA; :: thesis: for I being really-closed Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_halting_on s1,P1 holds
I is_halting_on s2,P2
let I be really-closed Program of SCM+FSA; :: thesis: ( DataPart s1 = DataPart s2 & I is_halting_on s1,P1 implies I is_halting_on s2,P2 )
set S1 = Initialize s1;
set S2 = Initialize s2;
IC (Initialize s1) = 0 by MEMSTR_0:47;
then A1: IC (Initialize s1) in dom I by AFINSQ_1:65;
A2: I c= P1 +* I by FUNCT_4:25;
defpred S₁[ Nat] means ( IC (Comput ((P1 +* I),(Initialize s1),$1)) = IC (Comput ((P2 +* I),(Initialize s2),$1)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),$1))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),$1))) & DataPart (Comput ((P1 +* I),(Initialize s1),$1)) = DataPart (Comput ((P2 +* I),(Initialize s2),$1)) );
A3: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
A4: IC (Comput ((P1 +* I),(Initialize s1),0)) = IC (Start-At (0,SCM+FSA)) by A3, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
A5: IC (Comput ((P2 +* I),(Initialize s2),0)) = IC (Start-At (0,SCM+FSA)) by A3, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
assume DataPart s1 = DataPart s2 ; :: thesis: ( not I is_halting_on s1,P1 or I is_halting_on s2,P2 )
then A6: Comput ((P1 +* I),(Initialize s1),0) = Comput ((P2 +* I),(Initialize s2),0) by MEMSTR_0:80;

A7: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
A8: Comput ((P2 +* I),(Initialize s2),(k + 1)) = Following ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k)))),(Comput ((P2 +* I),(Initialize s2),k))) ;
assume A9: S₁[k] ; :: thesis: S₁[k + 1]
then A10: for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f by SCMFSA_M:2;
for a being Int-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . a = (Comput ((P2 +* I),(Initialize s2),k)) . a by A9, SCMFSA_M:2;
then A11: Comput ((P1 +* I),(Initialize s1),k) = Comput ((P2 +* I),(Initialize s2),k) by A9, A10, SCMFSA_2:61;
A12: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) in dom I by A1, A2, AMISTD_1:21;
Comput ((P1 +* I),(Initialize s1),(k + 1)) = Following ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k)))),(Comput ((P1 +* I),(Initialize s1),k))) ;
then A13: Comput ((P1 +* I),(Initialize s1),(k + 1)) = Comput ((P2 +* I),(Initialize s2),(k + 1)) by A9, A11, A8;
A14: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* I),(Initialize s2),(k + 1))) by A13;
A15: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by PBOOLE:143;
A16: (P2 +* I) /. (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) = (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) by PBOOLE:143;
A17: I c= P1 +* I by FUNCT_4:25;
A18: I c= P2 +* I by FUNCT_4:25;
CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),(k + 1)))) = I . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by A12, A15, A17, GRFUNC_1:2
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),(k + 1)))) by A14, A12, A16, A18, GRFUNC_1:2 ;
hence S₁[k + 1] by A13; :: thesis: verum

end;

assume I is_halting_on s1,P1 ; :: thesis: I is_halting_on s2,P2
then P1 +* I halts_on Initialize s1 by SCMFSA7B:def 7;
then consider m being Nat such that
A19: CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),m))) = halt SCM+FSA ;
A20: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),0))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),0))) by PBOOLE:143;
A21: (P2 +* I) /. (IC (Comput ((P2 +* I),(Initialize s2),0))) = (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),0))) by PBOOLE:143;
A22: 0 in dom I by AFINSQ_1:65;
then CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),0))) = I . 0 by A4, A20, FUNCT_4:13
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),0))) by A5, A22, A21, FUNCT_4:13 ;
then A23: S₁[ 0 ] by A6;
for k being Nat holds S₁[k] from NAT_1:sch 2(A23, A7);
then CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),m))) = halt SCM+FSA by A19;
then P2 +* I halts_on Initialize s2 by EXTPRO_1:29;
hence I is_halting_on s2,P2 by SCMFSA7B:def 7; :: thesis: verum