let f1, f2 be Function of the carrier of X,REAL; :: thesis: ( ( for x being Point of X holds f1 . x = ||.x.|| ) & ( for x being Point of X holds f2 . x = ||.x.|| ) implies f1 = f2 )

assume that

A1: for x being Point of X holds f1 . x = ||.x.|| and

A2: for x being Point of X holds f2 . x = ||.x.|| ; :: thesis: f1 = f2

assume that

A1: for x being Point of X holds f1 . x = ||.x.|| and

A2: for x being Point of X holds f2 . x = ||.x.|| ; :: thesis: f1 = f2

now :: thesis: for x being Element of the carrier of X holds f1 . x = f2 . x

hence
f1 = f2
; :: thesis: verumlet x be Element of the carrier of X; :: thesis: f1 . x = f2 . x

thus f1 . x = ||.x.|| by A1

.= f2 . x by A2 ; :: thesis: verum

end;thus f1 . x = ||.x.|| by A1

.= f2 . x by A2 ; :: thesis: verum