let f1, f2 be Function of REAL ,REAL ; :: thesis: ( ( for x being Real holds f1 . x = sgn x ) & ( for x being Real holds f2 . x = sgn x ) implies f1 = f2 )
assume A6: for x being Real holds f1 . x = sgn x ; :: thesis: ( ex x being Real st not f2 . x = sgn x or f1 = f2 )
assume A7: for x being Real holds f2 . x = sgn x ; :: thesis: f1 = f2
for x being Real holds f1 . x = f2 . x

proof

let x be Real; :: thesis: f1 . x = f2 . x
thus f1 . x = sgn x by A6
.= f2 . x by A7 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:113; :: thesis: verum