let f, g be Action of (SCM-Instr R),(product ((SCM-VAL R) * SCM-OK)); :: thesis: ( ( for x being Element of SCM-Instr R

for y being SCM-State of R holds (f . x) . y = SCM-Exec-Res (x,y) ) & ( for x being Element of SCM-Instr R

for y being SCM-State of R holds (g . x) . y = SCM-Exec-Res (x,y) ) implies f = g )

assume that

A2: for x being Element of SCM-Instr R

for y being SCM-State of R holds (f . x) . y = SCM-Exec-Res (x,y) and

A3: for x being Element of SCM-Instr R

for y being SCM-State of R holds (g . x) . y = SCM-Exec-Res (x,y) ; :: thesis: f = g

for y being SCM-State of R holds (f . x) . y = SCM-Exec-Res (x,y) ) & ( for x being Element of SCM-Instr R

for y being SCM-State of R holds (g . x) . y = SCM-Exec-Res (x,y) ) implies f = g )

assume that

A2: for x being Element of SCM-Instr R

for y being SCM-State of R holds (f . x) . y = SCM-Exec-Res (x,y) and

A3: for x being Element of SCM-Instr R

for y being SCM-State of R holds (g . x) . y = SCM-Exec-Res (x,y) ; :: thesis: f = g

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

hence
f = g
by FUNCT_2:63; :: thesis: verumlet x be Element of SCM-Instr R; :: thesis: f . x = g . x

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

end;reconsider gx = g . x, fx = f . x as Function of (product ((SCM-VAL R) * SCM-OK)),(product ((SCM-VAL R) * SCM-OK)) by FUNCT_2:66;

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

hence
f . x = g . x
by FUNCT_2:63; :: thesis: verumlet y be SCM-State of R; :: thesis: fx . y = gx . y

thus fx . y = SCM-Exec-Res (x,y) by A2

.= gx . y by A3 ; :: thesis: verum

end;thus fx . y = SCM-Exec-Res (x,y) by A2

.= gx . y by A3 ; :: thesis: verum