let f1, f2 be Function of Omega,REAL; :: thesis: ( ( for w being Element of Omega holds
( ( ((Omega --> K) - RV) . w >= 0 implies f1 . w = ((Omega --> K) - RV) . w ) & ( ((Omega --> K) - RV) . w < 0 implies f1 . w = 0 ) ) ) & ( for w being Element of Omega holds
( ( ((Omega --> K) - RV) . w >= 0 implies f2 . w = ((Omega --> K) - RV) . w ) & ( ((Omega --> K) - RV) . w < 0 implies f2 . w = 0 ) ) ) implies f1 = f2 )
assume that
A2: for w being Element of Omega holds
( ( ((Omega --> K) - RV) . w >= 0 implies f1 . w = ((Omega --> K) - RV) . w ) & ( ((Omega --> K) - RV) . w < 0 implies f1 . w = 0 ) ) and
A3: for w being Element of Omega holds
( ( ((Omega --> K) - RV) . w >= 0 implies f2 . w = ((Omega --> K) - RV) . w ) & ( ((Omega --> K) - RV) . w < 0 implies f2 . w = 0 ) ) ; :: thesis: f1 = f2
let w be Element of Omega; :: according to FUNCT_2:def 8 :: thesis: f1 . w = f2 . w

per cases ( ((Omega --> K) - RV) . w >= 0 or ((Omega --> K) - RV) . w < 0 ) ;

suppose A0: ((Omega --> K) - RV) . w >= 0 ; :: thesis: f1 . w = f2 . w

then f1 . w = ((Omega --> K) - RV) . w by A2;
hence f1 . w = f2 . w by A0, A3; :: thesis: verum

end;

suppose A0: ((Omega --> K) - RV) . w < 0 ; :: thesis: f1 . w = f2 . w

then f1 . w = 0 by A2;
hence f1 . w = f2 . w by A0, A3; :: thesis: verum

end;

end;