reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;

theorem Th1:
  for F be Function-yielding Function, a be object holds
     a in Values F
  iff
     ex x,y st x in dom F & y in dom (F.x) & a = F.x.y
proof
  let F be Function-yielding Function, a be object;
  A1:Values F = Union rngs F by MATRIX_0:def 9
             .= union rng rngs F by CARD_3:def 4;
  A2:dom rngs F = dom F by FUNCT_6:def 3;
  thus a in Values F implies ex x,y be object st x in dom F & y in dom (F.x) &
    a = F.x.y
  proof
    assume a in Values F;
    then consider y be set such that
    A3: a in y & y in rng rngs F by A1,TARSKI:def 4;
    consider z be object such that
    A4: z in dom rngs F & (rngs F).z = y by A3,FUNCT_1:def 3;
    y = rng (F.z) by A2,A4,FUNCT_6:def 3;
    then ex x be object st x in dom (F.z) & a=(F.z).x by A3,FUNCT_1:def 3;
    hence thesis by A2,A4;
  end;
  given x,y be object such that
  A5: x in dom F & y in dom (F.x) & a = F.x.y;
  (rngs F).x = rng (F.x) by A5,FUNCT_6:def 3;
  then A6:rng (F.x) in rng rngs F by A5,A2,FUNCT_1:def 3;
  a in rng (F.x) by A5,FUNCT_1:def 3;
  hence thesis by A6,TARSKI:def 4,A1;
end;
