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 A2: for x being Real holds f1 . x = sgn x ; :: thesis: ( ex x being Real st not f2 . x = sgn x or f1 = f2 )

assume A3: for x being Real holds f2 . x = sgn x ; :: thesis: f1 = f2

for x being Element of REAL holds f1 . x = f2 . x

assume A2: for x being Real holds f1 . x = sgn x ; :: thesis: ( ex x being Real st not f2 . x = sgn x or f1 = f2 )

assume A3: for x being Real holds f2 . x = sgn x ; :: thesis: f1 = f2

for x being Element of REAL holds f1 . x = f2 . x

proof

hence
f1 = f2
by FUNCT_2:63; :: thesis: verum
let x be Element of REAL ; :: thesis: f1 . x = f2 . x

thus f1 . x = sgn x by A2

.= f2 . x by A3 ; :: thesis: verum

end;thus f1 . x = sgn x by A2

.= f2 . x by A3 ; :: thesis: verum