reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;

theorem Th37:
  for B being permutation of A holds dom B = dom A & rng B = rng A
  proof
    let B be permutation of A;
    consider f being Permutation of dom A such that
A1: B = A*f by Def9;
    dom A <> {} implies dom A <> {}; then
    rng f = dom A & dom f = dom A by FUNCT_2:def 1,def 3;
    hence thesis by A1,RELAT_1:27,28;
  end;
