let q be NAT -defined the Instructions 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 Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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 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 Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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+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 Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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: ( 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 Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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 Instruction-Sequence of SCM+FSA; :: thesis: ( q c= P1 & q c= P2 implies for i being Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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 Int-Location
for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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 Int-Location ; :: thesis: for loc being Element of NAT st CurInstr (P1,(Comput (P1,s1,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: ( CurInstr (P1,(Comput (P1,s1,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 ) )
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)) . (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;
assume that
A6: CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc and
A7: loc <> succ (IC (Comput (P1,s1,i))) ; :: 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 B1, A2, AMISTD_5:7;

A9: 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 A8, A3, A4, A6, SCMFSA_2:70;
hence contradiction by B1, 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 B1, A2, EXTPRO_1:def 10;

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 A8, A3, A4, A6, SCMFSA_2:70;
hence contradiction by B1, 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