set X = the carrier of T;

let f1, f2 be Function of (bool the carrier of T),(bool the carrier of T); :: thesis: ( ( for X being Subset of T holds f1 . X = Cl X ) & ( for X being Subset of T holds f2 . X = Cl X ) implies f1 = f2 )

assume that

A2: for x being Subset of the carrier of T holds f1 . x = Cl x and

A3: for x being Subset of the carrier of T holds f2 . x = Cl x ; :: thesis: f1 = f2

let x be Subset of the carrier of T; :: according to FUNCT_2:def 8 :: thesis: f1 . x = f2 . x

thus f1 . x = Cl x by A2

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

let f1, f2 be Function of (bool the carrier of T),(bool the carrier of T); :: thesis: ( ( for X being Subset of T holds f1 . X = Cl X ) & ( for X being Subset of T holds f2 . X = Cl X ) implies f1 = f2 )

assume that

A2: for x being Subset of the carrier of T holds f1 . x = Cl x and

A3: for x being Subset of the carrier of T holds f2 . x = Cl x ; :: thesis: f1 = f2

let x be Subset of the carrier of T; :: according to FUNCT_2:def 8 :: thesis: f1 . x = f2 . x

thus f1 . x = Cl x by A2

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