
theorem Th5:
  for X be non empty set, F be Domain-Sequence,
      f,g being Function of [:X,product F:],product F st
  for x be Element of X, d being Element of product F,
      i being Element of dom F holds (f.(x,d)).i = (g.(x,d)).i holds f = g
proof
  let X be non empty set;
  let F be Domain-Sequence;
  let f,g being Function of [:X,product F:],product F such that
A1: for x be Element of X, d being Element of product F, i being Element
  of dom F holds (f.(x,d)).i = (g.(x,d)).i;
  now
    let x be Element of X,d being Element of product F;
A2: dom(f.(x,d)) = dom F & dom(g.(x,d)) = dom F by CARD_3:9;
    for v be object st v in dom F holds (f.(x,d)).v = (g.(x,d)).v by A1;
    hence f.(x,d) = g.(x,d) by A2,FUNCT_1:2;
  end;
  hence thesis by BINOP_1:2;
end;
