reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;
reserve Y for non empty set,
  f for Function of X,Y,
  p for PartFunc of Y,Z,
  x for Element of X;
reserve g for Function of X,X;

theorem
  rng f c= dom p implies (p/*f)*g = p/*(f*g)
proof
A1: rng(f*g) c= rng f by RELAT_1:26;
  assume
A2: rng f c= dom p;
  hence (p/*f)*g = p*f*g by Def11
    .= p*(f*g) by RELAT_1:36
    .= p /* (f*g) by A2,A1,Def11,XBOOLE_1:1;
end;
