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
A1: for x being Element of REAL n holds f . x = x . i and
A2: for x being Element of REAL n holds g . x = x . i ; :: thesis: f = g

now

let x be Element of REAL n; :: thesis: f . x = g . x
f . x = x . i by A1;
hence f . x = g . x by A2; :: thesis: verum

end;

hence f = g by FUNCT_2:63; :: thesis: verum