let f1, f2 be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f1 . x = F₁(x) ) & ( for x being Real holds f2 . x = F₁(x) ) implies f1 = f2 )
assume that
A1: for x being Real holds f1 . x = F₁(x) and
A2: for x being Real holds f2 . x = F₁(x) ; :: thesis: f1 = f2

now :: thesis: for x being Element of REAL holds f1 . x = f2 . x

let x be Element of REAL ; :: thesis: f1 . x = f2 . x
thus f1 . x = F₁(x) by A1
.= f2 . x by A2 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:63; :: thesis: verum