let q be NAT -defined the InstructionsF of SCM+FSA -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )

let p be non empty q -autonomic FinPartState of SCM+FSA; :: thesis: for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )

let s1, s2 be State of SCM+FSA; :: thesis: ( p c= s1 & p c= s2 implies for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )

assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )

let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: ( q c= P1 & q c= P2 implies for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )

assume A2: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )

let i be Nat; :: thesis: for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )

let da be Int-Location; :: thesis: for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )

let loc be Nat; :: thesis: ( CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 implies ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
set I = CurInstr (P1,(Comput (P1,s1,i)));
set Cs1i = Comput (P1,s1,i);
set Cs2i = Comput (P2,s2,i);
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i1 = Comput (P2,s2,(i + 1));
A3: Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
A4: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
IC in dom p by AMISTD_5:6;
then A5: ( ((Comput (P1,s1,(i + 1))) | (dom p)) . () = (Comput (P1,s1,(i + 1))) . () & ((Comput (P2,s2,(i + 1))) | (dom p)) . () = (Comput (P2,s2,(i + 1))) . () ) by FUNCT_1:49;
assume that
A6: CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc and
A7: loc <> (IC (Comput (P1,s1,i))) + 1 ; :: thesis: ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
A8: CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) by ;
A9: now :: thesis: ( (Comput (P2,s2,i)) . da = 0 implies not (Comput (P1,s1,i)) . da <> 0 )
assume ( (Comput (P2,s2,i)) . da = 0 & (Comput (P1,s1,i)) . da <> 0 ) ; :: thesis: contradiction
then ( (Comput (P2,s2,(i + 1))) . () = loc & (Comput (P1,s1,(i + 1))) . () = (IC (Comput (P1,s1,i))) + 1 ) by ;
hence contradiction by A1, A5, A7, A2, EXTPRO_1:def 10; :: thesis: verum
end;
A10: (Comput (P1,s1,(i + 1))) | (dom p) = (Comput (P2,s2,(i + 1))) | (dom p) by ;
now :: thesis: ( (Comput (P1,s1,i)) . da = 0 implies not (Comput (P2,s2,i)) . da <> 0 )
assume ( (Comput (P1,s1,i)) . da = 0 & (Comput (P2,s2,i)) . da <> 0 ) ; :: thesis: contradiction
then ( (Comput (P1,s1,(i + 1))) . () = loc & (Comput (P2,s2,(i + 1))) . () = (IC (Comput (P2,s2,i))) + 1 ) by ;
hence contradiction by A1, A5, A10, A7, A2, AMISTD_5:7; :: thesis: verum
end;
hence ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) by A9; :: thesis: verum