let f, g be Function of (REAL n),REAL; :: thesis: ( ( for x being Element of REAL n holds f . x = x . i ) & ( for x being Element of REAL n holds g . x = x . i ) implies f = g )

assume that

A2: for x being Element of REAL n holds f . x = x . i and

A3: for x being Element of REAL n holds g . x = x . i ; :: thesis: f = g

assume that

A2: for x being Element of REAL n holds f . x = x . i and

A3: for x being Element of REAL n holds g . x = x . i ; :: thesis: f = g

now :: thesis: for x being Element of REAL n holds f . x = g . x

hence
f = g
by FUNCT_2:63; :: thesis: verumlet x be Element of REAL n; :: thesis: f . x = g . x

f . x = x . i by A2;

hence f . x = g . x by A3; :: thesis: verum

end;f . x = x . i by A2;

hence f . x = g . x by A3; :: thesis: verum