reserve X,Y for set;
reserve G for Group;
reserve n for Nat;

theorem
  X,Y are_equipotent implies SymGroup(X),SymGroup(Y) are_isomorphic
  proof
    assume
A1: X,Y are_equipotent;
    then consider p being Function such that
A2: p is one-to-one and
A3: dom p = X and
A4: rng p = Y by WELLORD2:def 4;
    per cases;
    suppose X = {};
      hence thesis by A1,CARD_1:26;
    end;
    suppose
A5:   X <> {};
      then
A6:   Y <> {} by A1,CARD_1:26;
      reconsider p as Function of X,Y by A3,A4,FUNCT_2:2;
A7:   p is onto by A4;
      then reconsider h = SymGroupsIso(p) as
      Homomorphism of SymGroup(X),SymGroup(Y) by A2,A5,A6,Th10;
      take h;
      thus h is one-to-one onto by A2,A5,A6,A7,Th11,Th12;
    end;
  end;
