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 Th40:
  A is permutation of B & B is permutation of C implies A is permutation of C
  proof assume
A1: A is permutation of B & B is permutation of C; then
A2: dom C = dom B by Th37;
    consider f being Permutation of dom B such that
A3: A = B*f by A1,Def9;
    consider g being Permutation of dom C such that
A4: B = C*g by A1,Def9;
    reconsider h = g*f as Permutation of dom C by A2;
    take h;
    thus A = C*h by A3,A4,RELAT_1:36;
  end;
