reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;
reserve f for Function;
reserve x1,x2,x3,x4,x5 for object;
reserve p for FinSequence;
reserve ND for non empty set;
reserve y1,y2,y3,y4,y5 for Element of ND;

theorem
  for S being set, D1, D2 being non empty set, f1 being Function of
  S, D1, f2 being Function of D1, D2 holds f1 is bijective & f2 is bijective
  implies f2*f1 is bijective
proof
  let S be set, D1, D2 be non empty set, f1 be Function of S, D1, f2 be
  Function of D1, D2;
  set f3 = f2*f1;
A1: dom f2 = D1 by FUNCT_2:def 1;
  assume
A2: f1 is bijective & f2 is bijective;
  then rng f2 = D2 & rng f1 = D1 by FUNCT_2:def 3;
  then rng f3 = D2 by A1,RELAT_1:28;
  then f3 is one-to-one onto by A2,FUNCT_2:def 3;
  hence thesis;
end;
