theorem Th54A:
  (for t st t in rng p holds t is x-omitting) iff o-term p is x-omitting
  proof
A0: now assume
A1:   for i st i in dom p holds p/.i is x-omitting;
      let t; assume t in rng p;
      then consider i being object such that
A2:   i in dom p & t = p.i by FUNCT_1:def 3;
      reconsider i as Nat by A2;
      t = p/.i by A2,PARTFUN1:def 6;
      hence t is x-omitting by A1,A2;
    end;
    now assume
A3:   for t st t in rng p holds t is x-omitting;
      let i; assume i in dom p;
      then p/.i = p.i in rng p by FUNCT_1:def 3,PARTFUN1:def 6;
      hence p/.i is x-omitting by A3;
    end;
    hence thesis by A0,Th54;
  end;
