let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = da =0_goto loc & loc <> succ (IC (Comput ((ProgramPart s1),s1,i))) holds
( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 )
let s1, s2 be State of SCM+FSA; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = da =0_goto loc & loc <> succ (IC (Comput ((ProgramPart s1),s1,i))) holds
( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 ) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = da =0_goto loc & loc <> succ (IC (Comput ((ProgramPart s1),s1,i))) holds
( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 )
let i be Element of NAT ; :: thesis: for da being Int-Location
for loc being Element of NAT st CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = da =0_goto loc & loc <> succ (IC (Comput ((ProgramPart s1),s1,i))) holds
( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 )
let da be Int-Location ; :: thesis: for loc being Element of NAT st CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = da =0_goto loc & loc <> succ (IC (Comput ((ProgramPart s1),s1,i))) holds
( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 )
let loc be Element of NAT ; :: thesis: ( CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = da =0_goto loc & loc <> succ (IC (Comput ((ProgramPart s1),s1,i))) implies ( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 ) )
set I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i)));
set Cs1i = Comput ((ProgramPart s1),s1,i);
set Cs2i = Comput ((ProgramPart s2),s2,i);
set Cs1i1 = Comput ((ProgramPart s1),s1,(i + 1));
set Cs2i1 = Comput ((ProgramPart s2),s2,(i + 1));
A2: Comput ((ProgramPart s1),s1,(i + 1)) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i)))),(Comput ((ProgramPart s1),s1,i))) ;
A3: Comput ((ProgramPart s2),s2,(i + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i)))),(Comput ((ProgramPart s2),s2,i))) ;
A4: ( ((Comput ((ProgramPart s1),s1,(i + 1))) | (dom p)) . (IC SCM+FSA) = (Comput ((ProgramPart s1),s1,(i + 1))) . (IC SCM+FSA) & ((Comput ((ProgramPart s2),s2,(i + 1))) | (dom p)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,(i + 1))) . (IC SCM+FSA) ) by Th15, FUNCT_1:72;
assume that
A5: CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = da =0_goto loc and
A6: loc <> succ (IC (Comput ((ProgramPart s1),s1,i))) ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 )
A7: CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) by A1, Th18;

A8: now

assume ( (Comput ((ProgramPart s2),s2,i)) . da = 0 & (Comput ((ProgramPart s1),s1,i)) . da <> 0 ) ; :: thesis: contradiction
then ( (Comput ((ProgramPart s2),s2,(i + 1))) . (IC SCM+FSA) = loc & (Comput ((ProgramPart s1),s1,(i + 1))) . (IC SCM+FSA) = succ (IC (Comput ((ProgramPart s1),s1,i))) ) by A7, A2, A3, A5, SCMFSA_2:96;
hence contradiction by A1, A4, A6, EXTPRO_1:def 9; :: thesis: verum

end;

A9: (Comput ((ProgramPart s1),s1,(i + 1))) | (dom p) = (Comput ((ProgramPart s2),s2,(i + 1))) | (dom p) by A1, EXTPRO_1:def 9;

now

assume ( (Comput ((ProgramPart s1),s1,i)) . da = 0 & (Comput ((ProgramPart s2),s2,i)) . da <> 0 ) ; :: thesis: contradiction
then ( (Comput ((ProgramPart s1),s1,(i + 1))) . (IC SCM+FSA) = loc & (Comput ((ProgramPart s2),s2,(i + 1))) . (IC SCM+FSA) = succ (IC (Comput ((ProgramPart s2),s2,i))) ) by A7, A2, A3, A5, SCMFSA_2:96;
hence contradiction by A1, A4, A9, A6, Th18; :: thesis: verum

end;

hence ( (Comput ((ProgramPart s1),s1,i)) . da = 0 iff (Comput ((ProgramPart s2),s2,i)) . da = 0 ) by A8; :: thesis: verum