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 Th39:
  A is permutation of B implies B is permutation of A
  proof assume
A1: A is permutation of B; then
A2: dom A = dom B by Th37;
    consider f being Permutation of dom B such that
A3: A = B*f by A1,Def9;
    reconsider g = f" as Permutation of dom A by A2;
    take g;
    thus B = B*id dom B by RELAT_1:52
    .= B*(f*g) by FUNCT_2:61
    .= A*g by A3,RELAT_1:36;
  end;
