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 s1),(Comput ((ProgramPart s1),s1,i))) holds
( IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) & I = CurInstr ((ProgramPart s2),(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 s1),(Comput ((ProgramPart s1),s1,i))) holds
( IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) & I = CurInstr ((ProgramPart s2),(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 s1),(Comput ((ProgramPart s1),s1,i))) holds
( IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) & I = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) )
let i be Element of NAT ; :: thesis: for I being Instruction of SCM st I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) holds
( IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) & I = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) )
let I be Instruction of SCM; :: thesis: ( I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) implies ( IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) & I = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) ) )
assume A3: I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) ; :: thesis: ( IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) & I = CurInstr ((ProgramPart s2),(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 s2),(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, EXTPRO_1:def 9; :: thesis: verum

end;

thus I = CurInstr ((ProgramPart s2),(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;
TX: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,i)) by AMI_1:123;
TY: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,i)) by AMI_1:123;
Y: (ProgramPart s2) /. (IC (Comput ((ProgramPart s2),s2,i))) = (Comput ((ProgramPart s2),s2,i)) . (IC (Comput ((ProgramPart s2),s2,i))) by TY, COMPOS_1:38;
Z: (ProgramPart s1) /. (IC (Comput ((ProgramPart s1),s1,i))) = (Comput ((ProgramPart s1),s1,i)) . (IC (Comput ((ProgramPart s1),s1,i))) by TX, COMPOS_1:38;
assume ZZ: I <> CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) ; :: thesis: contradiction
(Comput ((ProgramPart s1),s1,i)) | (dom p) = (Comput ((ProgramPart s2),s2,i)) | (dom p) by A1, A2, EXTPRO_1:def 9;
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;