let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for i being Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) holds
( IC (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) & I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) )
let s1, s2 be State of SCM ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) holds
( IC (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) & I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) ) )
assume that
A1: p c= s1 and
A2: p c= s2 ; :: thesis: for i being Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) holds
( IC (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) & I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) )
let i be Element of NAT ; :: thesis: for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) holds
( IC (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) & I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) )
let I be Instruction of SCM ; :: thesis: ( I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) implies ( IC (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) & I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) ) )
assume A3: I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) ; :: thesis: ( IC (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) & I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) )
set Cs2i = Comput (ProgramPart s2),s2,i;
set Cs1i = Comput (ProgramPart s1),s1,i;
thus A4: IC (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) :: thesis: I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i)

proof

assume A5: IC (Comput (ProgramPart s1),s1,i) <> IC (Comput (ProgramPart s2),s2,i) ; :: thesis: contradiction
( ((Comput (ProgramPart s1),s1,i) | (dom p)) . (IC SCM ) = (Comput (ProgramPart s1),s1,i) . (IC SCM ) & ((Comput (ProgramPart s2),s2,i) | (dom p)) . (IC SCM ) = (Comput (ProgramPart s2),s2,i) . (IC SCM ) ) by Th84, FUNCT_1:72;
hence contradiction by A1, A2, A5, AMI_1:def 25; :: thesis: verum

end;

thus I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) :: thesis: verum

proof

ProgramPart p c= p by RELAT_1:88;
then A6: dom (ProgramPart p) c= dom p by GRFUNC_1:8;
IC (Comput (ProgramPart s2),s2,i) in dom (ProgramPart p) by A2, Th86;
then A7: ((Comput (ProgramPart s2),s2,i) | (dom p)) . (IC (Comput (ProgramPart s2),s2,i)) = (Comput (ProgramPart s2),s2,i) . (IC (Comput (ProgramPart s2),s2,i)) by A6, FUNCT_1:72;
IC (Comput (ProgramPart s1),s1,i) in dom (ProgramPart p) by A1, Th86;
then A8: ((Comput (ProgramPart s1),s1,i) | (dom p)) . (IC (Comput (ProgramPart s1),s1,i)) = (Comput (ProgramPart s1),s1,i) . (IC (Comput (ProgramPart s1),s1,i)) by A6, FUNCT_1:72;
Y: (ProgramPart (Comput (ProgramPart s2),s2,i)) /. (IC (Comput (ProgramPart s2),s2,i)) = (Comput (ProgramPart s2),s2,i) . (IC (Comput (ProgramPart s2),s2,i)) by AMI_1:150;
Z: (ProgramPart (Comput (ProgramPart s1),s1,i)) /. (IC (Comput (ProgramPart s1),s1,i)) = (Comput (ProgramPart s1),s1,i) . (IC (Comput (ProgramPart s1),s1,i)) by AMI_1:150;
assume ZZ: I <> CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) ; :: thesis: contradiction
(Comput (ProgramPart s1),s1,i) | (dom p) = (Comput (ProgramPart s2),s2,i) | (dom p) by A1, A2, AMI_1:def 25;
then ((Comput (ProgramPart s1),s1,i) | (dom p)) . (IC (Comput (ProgramPart s1),s1,i)) = (Comput (ProgramPart s2),s2,i) . (IC (Comput (ProgramPart s1),s1,i)) by A7, A4;
hence contradiction by A3, A4, A8, Y, ZZ, Z; :: thesis: verum

end;