let p be non NAT -defined autonomic FinPartState of SCM+FSA ; :: thesis: for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT holds
( IC (Computation s1,i) = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) )
let s1, s2 be State of SCM+FSA ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT holds
( IC (Computation s1,i) = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) ) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT holds
( IC (Computation s1,i) = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) )
let i be Element of NAT ; :: thesis: ( IC (Computation s1,i) = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) )
set I = CurInstr (Computation s1,i);
set Cs1i = Computation s1,i;
set Cs2i = Computation s2,i;
thus A2: IC (Computation s1,i) = IC (Computation s2,i) :: thesis: CurInstr (Computation s1,i) = CurInstr (Computation s2,i)

proof

assume A3: IC (Computation s1,i) <> IC (Computation s2,i) ; :: thesis: contradiction
( ((Computation s1,i) | (dom p)) . (IC SCM+FSA ) = (Computation s1,i) . (IC SCM+FSA ) & ((Computation s2,i) | (dom p)) . (IC SCM+FSA ) = (Computation s2,i) . (IC SCM+FSA ) ) by Th15, FUNCT_1:72;
hence contradiction by A1, A3, AMI_1:def 25; :: thesis: verum

end;

thus CurInstr (Computation s1,i) = CurInstr (Computation s2,i) :: thesis: verum

proof

assume A4: CurInstr (Computation s1,i) <> CurInstr (Computation s2,i) ; :: thesis: contradiction
A5: ( IC (Computation s1,i) in dom (ProgramPart p) & IC (Computation s2,i) in dom (ProgramPart p) ) by A1, Th17;
ProgramPart p c= p by RELAT_1:88;
then dom (ProgramPart p) c= dom p by GRFUNC_1:8;
then ( ((Computation s1,i) | (dom p)) . (IC (Computation s1,i)) = (Computation s1,i) . (IC (Computation s1,i)) & ((Computation s2,i) | (dom p)) . (IC (Computation s2,i)) = (Computation s2,i) . (IC (Computation s2,i)) ) by A5, FUNCT_1:72;
hence contradiction by A1, A2, A4, AMI_1:def 25; :: thesis: verum

end;