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 Th112:
  for A,B being set, f being Function of A,B st y in rng f ex x
  being Element of A st y = f.x
proof
  let A,B be set, f be Function of A,B;
  assume y in rng f;
  then consider x being object such that
A1: x in A and
A2: f.x = y by Th11;
  reconsider x as Element of A by A1;
  take x;
  thus thesis by A2;
end;
