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 Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i1 = Comput (P2,s2,(i + 1));
A3: (Comput (P1,s1,(i + 1))) | (dom p) = (Comput (P2,s2,(i + 1))) | (dom p) by A1, A2, EXTPRO_1:def 10;
set Cs2i = Comput (P2,s2,i);
set Cs1i = Comput (P1,s1,i);
set I = CurInstr (P1,(Comput (P1,s1,i)));
A4: 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))) ;
IC in dom p by AMISTD_5:6;
then A5: ( ((Comput (P1,s1,(i + 1))) | (dom p)) . (IC ) = (Comput (P1,s1,(i + 1))) . (IC ) & ((Comput (P2,s2,(i + 1))) | (dom p)) . (IC ) = (Comput (P2,s2,(i + 1))) . (IC ) ) by FUNCT_1:49;
A6: 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))) ;
assume that
A7: CurInstr (P1,(Comput (P1,s1,i))) = da >0_goto loc and
A8: loc <> (IC (Comput (P1,s1,i))) + 1 ; :: thesis: ( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
A9: CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) by A1, A2, AMISTD_5:7;

A10: now :: thesis: ( (Comput (P2,s2,i)) . da > 0 implies not (Comput (P1,s1,i)) . da <= 0 )

assume that
A11: (Comput (P2,s2,i)) . da > 0 and
A12: (Comput (P1,s1,i)) . da <= 0 ; :: thesis: contradiction
(Comput (P2,s2,(i + 1))) . (IC ) = loc by A9, A6, A7, A11, SCMFSA_2:71;
hence contradiction by A4, A5, A3, A7, A8, A12, SCMFSA_2:71; :: thesis: verum

end;

A13: IC (Comput (P1,s1,i)) = IC (Comput (P2,s2,i)) by A1, A2, AMISTD_5:7;

now :: thesis: ( (Comput (P1,s1,i)) . da > 0 implies not (Comput (P2,s2,i)) . da <= 0 )

assume that
A14: (Comput (P1,s1,i)) . da > 0 and
A15: (Comput (P2,s2,i)) . da <= 0 ; :: thesis: contradiction
(Comput (P1,s1,(i + 1))) . (IC ) = loc by A4, A7, A14, SCMFSA_2:71;
hence contradiction by A13, A9, A6, A5, A3, A7, A8, A15, SCMFSA_2:71; :: thesis: verum

end;

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