let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM st NPP p c= s1 & NPP p c= s2 holds
for P1, P2 being the Instructions of SCM -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let s1, s2 be State of SCM; :: thesis: ( NPP p c= s1 & NPP p c= s2 implies for P1, P2 being the Instructions of SCM -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
assume B1: ( NPP p c= s1 & NPP p c= s2 ) ; :: thesis: for P1, P2 being the Instructions of SCM -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let P1, P2 be the Instructions of SCM -valued ManySortedSet of NAT ; :: thesis: ( ProgramPart p c= P1 & ProgramPart p c= P2 implies for i being Element of NAT
for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
assume A2: ( ProgramPart p c= P1 & ProgramPart p c= P2 ) ; :: thesis: for i being Element of NAT
for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let i be Element of NAT ; :: thesis: for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let da be Data-Location ; :: thesis: for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let loc be Element of NAT ; :: thesis: for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let I be Instruction of SCM; :: thesis: ( I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) implies ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
assume A3: I = CurInstr (P1,(Comput (P1,s1,i))) ; :: thesis: ( not I = da =0_goto loc or not loc <> succ (IC (Comput (P1,s1,i))) or ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
set Cs2i1 = Comput (P2,s2,(i + 1));
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i = Comput (P2,s2,i);
set Cs1i = Comput (P1,s1,i);
A4: Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:4
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
A5: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:4
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
IC in dom p by AMISTD_5:6;
then IC in dom (NPP p) by COMPOS_1:179;
then A6: ( ((Comput (P1,s1,(i + 1))) | (dom (NPP p))) . (IC ) = (Comput (P1,s1,(i + 1))) . (IC ) & ((Comput (P2,s2,(i + 1))) | (dom (NPP p))) . (IC ) = (Comput (P2,s2,(i + 1))) . (IC ) ) by FUNCT_1:72;
assume that
A7: I = da =0_goto loc and
A8: loc <> succ (IC (Comput (P1,s1,i))) ; :: thesis: ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
A9: I = CurInstr (P2,(Comput (P2,s2,i))) by A3, A2, B1, AMISTD_5:7;

A10: now

assume ( (Comput (P2,s2,i)) . da = 0 & (Comput (P1,s1,i)) . da <> 0 ) ; :: thesis: contradiction
then ( (Comput (P2,s2,(i + 1))) . (IC ) = loc & (Comput (P1,s1,(i + 1))) . (IC ) = succ (IC (Comput (P1,s1,i))) ) by A3, A9, A4, A5, A7, AMI_3:14;
hence contradiction by A6, A8, A2, B1, EXTPRO_1:def 9; :: thesis: verum

end;

A11: (Comput (P1,s1,(i + 1))) | (dom (NPP p)) = (Comput (P2,s2,(i + 1))) | (dom (NPP p)) by A2, B1, EXTPRO_1:def 9;

now

assume ( (Comput (P1,s1,i)) . da = 0 & (Comput (P2,s2,i)) . da <> 0 ) ; :: thesis: contradiction
then ( (Comput (P1,s1,(i + 1))) . (IC ) = loc & (Comput (P2,s2,(i + 1))) . (IC ) = succ (IC (Comput (P2,s2,i))) ) by A3, A9, A4, A5, A7, AMI_3:14;
hence contradiction by A6, A11, A8, A2, B1, AMISTD_5:7; :: thesis: verum

end;

hence ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) by A10; :: thesis: verum