let f, g be Function of SCM+FSA-Instr,(Funcs ((product (SCM*-VAL * SCM+FSA-OK)),(product (SCM*-VAL * SCM+FSA-OK)))); :: thesis: ( ( for x being Element of SCM+FSA-Instr
for y being SCM+FSA-State holds (f . x) . y = SCM+FSA-Exec-Res (x,y) ) & ( for x being Element of SCM+FSA-Instr
for y being SCM+FSA-State holds (g . x) . y = SCM+FSA-Exec-Res (x,y) ) implies f = g )
assume that
A2: for x being Element of SCM+FSA-Instr
for y being SCM+FSA-State holds (f . x) . y = SCM+FSA-Exec-Res (x,y) and
A3: for x being Element of SCM+FSA-Instr
for y being SCM+FSA-State holds (g . x) . y = SCM+FSA-Exec-Res (x,y) ; :: thesis: f = g
now :: thesis: for x being Element of SCM+FSA-Instr holds f . x = g . x
let x be Element of SCM+FSA-Instr ; :: thesis: f . x = g . x
reconsider gx = g . x, fx = f . x as Function of (product (SCM*-VAL * SCM+FSA-OK)),(product (SCM*-VAL * SCM+FSA-OK)) ;
now :: thesis: for y being SCM+FSA-State holds fx . y = gx . y
let y be SCM+FSA-State; :: thesis: fx . y = gx . y
thus fx . y = SCM+FSA-Exec-Res (x,y) by A2
.= gx . y by A3 ; :: thesis: verum
end;
hence f . x = g . x by FUNCT_2:63; :: thesis: verum
end;
hence f = g by FUNCT_2:63; :: thesis: verum