let f, g be Action of SCM-Instr,(product (SCM-VAL * SCM-OK)); :: thesis: ( ( for x being Element of SCM-Instr
for y being SCM-State holds (f . x) . y = SCM-Exec-Res (x,y) ) & ( for x being Element of SCM-Instr
for y being SCM-State holds (g . x) . y = SCM-Exec-Res (x,y) ) implies f = g )
assume that
A2: for x being Element of SCM-Instr
for y being SCM-State holds (f . x) . y = SCM-Exec-Res (x,y) and
A3: for x being Element of SCM-Instr
for y being SCM-State holds (g . x) . y = SCM-Exec-Res (x,y) ; :: thesis: f = g

now :: thesis: for x being Element of SCM-Instr holds f . x = g . x

let x be Element of SCM-Instr ; :: thesis: f . x = g . x
reconsider gx = g . x, fx = f . x as Function of (product (SCM-VAL * SCM-OK)),(product (SCM-VAL * SCM-OK)) by FUNCT_2:66;

now :: thesis: for y being SCM-State holds fx . y = gx . y

let y be SCM-State; :: thesis: fx . y = gx . y
thus fx . y = SCM-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