let f1, f2 be Function of (Tdisk (o,r)),(Tcircle (o,r)); :: thesis: ( ( for x being Point of (Tdisk (o,r)) holds f1 . x = HC (x,f) ) & ( for x being Point of (Tdisk (o,r)) holds f2 . x = HC (x,f) ) implies f1 = f2 )
assume that
A2: for x being Point of (Tdisk (o,r)) holds f1 . x = HC (x,f) and
A3: for x being Point of (Tdisk (o,r)) holds f2 . x = HC (x,f) ; :: thesis: f1 = f2
for x being Point of (Tdisk (o,r)) holds f1 . x = f2 . x

proof

let x be Point of (Tdisk (o,r)); :: thesis: f1 . x = f2 . x
thus f1 . x = HC (x,f) by A2
.= f2 . x by A3 ; :: thesis: verum

end;

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