reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem :: WAYBEL29:1
  for S, T being non empty 1-sorted for f being Function of S, T st f is
  one-to-one onto holds f*f" = id T & f"*f = id S & f" is one-to-one onto
proof
  let S, T be non empty 1-sorted;
  let f be Function of S, T;
A1: [#]T = the carrier of T;
  assume
A2: f is one-to-one onto;
  then
A3: rng f = the carrier of T by FUNCT_2:def 3;
  then dom f = the carrier of S & rng (f") = [#]S by A2,A1,FUNCT_2:def 1
,TOPS_2:49;
  hence thesis by A2,A3,A1,FUNCT_2:def 3,TOPS_2:50,52;
end;
