theorem
  U1,U2 are_isomorphic implies U2,U1 are_isomorphic
proof
  assume U1,U2 are_isomorphic;
  then consider f such that
A1: f is_isomorphism;
  f is_monomorphism by A1;
  then
A2: f is one-to-one;
  then
A3: rng(f") = dom f by FUNCT_1:33
    .= the carrier of U1 by FUNCT_2:def 1;
A4: f is_epimorphism by A1;
  dom(f") = rng f by A2,FUNCT_1:33
    .= the carrier of U2 by A4;
  then reconsider g = f" as Function of U2,U1 by A3,FUNCT_2:def 1,RELSET_1:4;
  take g;
  thus thesis by A1,Th10;
end;
