reserve A, B, X, Y for set;

theorem
  for A being Subset of X holds (delta X)"[:A,A:] = A
proof
  let A be Subset of X;
  set f = delta X;
  thus f"[:A,A:] c= A by Th2;
  let x be object;
  assume
A1: x in A;
  then f.x = [x,x] by FUNCT_3:def 6;
  then dom f = X & f.x in [:A,A:] by A1,FUNCT_3:def 6,ZFMISC_1:87;
  hence thesis by A1,FUNCT_1:def 7;
end;
