
theorem Th4:
  for X be set, F be FinSequence of X holds F is disjoint_valued
  iff for i,j be Nat st i in dom F & j in dom F & i <> j holds F.i misses F.j
proof
  let X be set;
  let F be FinSequence of X;
  now
    assume
A1: for i,j be Nat st i in dom F & j in dom F & i <> j holds F.i misses F.j;
    for x,y be object st x <> y holds F.x misses F.y
    proof
      let x,y be object;
      assume
A2:   x <> y;
      per cases;
      suppose
        x in dom F & y in dom F;
        hence thesis by A1,A2;
      end;
      suppose
        not x in dom F;
        then F.x = {} by FUNCT_1:def 2;
        hence thesis;
      end;
      suppose
        not y in dom F;
        then F.y = {} by FUNCT_1:def 2;
        hence thesis;
      end;
    end;
    hence F is disjoint_valued by PROB_2:def 2;
  end;
  hence thesis by PROB_2:def 2;
end;
