
theorem
  for f being Function st f is union-distributive for x,y being set st x
  in dom f & y in dom f & x \/ y in dom f holds f.(x \/ y) = (f.x) \/ (f.y)
proof
  let f be Function such that
A1: f is union-distributive;
  let x,y be set;
  set X = {x,y};
  assume that
A2: x in dom f & y in dom f and
A3: x \/ y in dom f;
A4: union X = x \/ y by ZFMISC_1:75;
  X c= dom f by A2,ZFMISC_1:32;
  hence f.(x \/ y) = union (f.:X) by A1,A3,A4
    .= union {f.x,f.y} by A2,FUNCT_1:60
    .= (f.x) \/ (f.y) by ZFMISC_1:75;
end;
