deffunc H₁( Subset of R, Subset of R) -> set = 1 - (JaccardIndex ($1,$2));
let f1, f2 be Function of [:(bool R),(bool R):],REAL; :: thesis: ( ( for A, B being Subset of R holds f1 . (A,B) = 1 - (JaccardIndex (A,B)) ) & ( for A, B being Subset of R holds f2 . (A,B) = 1 - (JaccardIndex (A,B)) ) implies f1 = f2 )
assume that
A1: for x, y being Subset of R holds f1 . (x,y) = H₁(x,y) and
A2: for x, y being Subset of R holds f2 . (x,y) = H₁(x,y) ; :: thesis: f1 = f2
for x, y being Subset of R holds f1 . (x,y) = f2 . (x,y)

proof

let x, y be Subset of R; :: thesis: f1 . (x,y) = f2 . (x,y)
f1 . (x,y) = H₁(x,y) by A1
.= f2 . (x,y) by A2 ;
hence f1 . (x,y) = f2 . (x,y) ; :: thesis: verum

end;

hence f1 = f2 by BINOP_1:2; :: thesis: verum