
theorem Th7:
  for i being object, J being ManySortedSet of {i} holds J = {i} --> J.i
proof
  let i be object, J be ManySortedSet of {i};
  A1: dom J = {i} by PARTFUN1:def 2
    .= dom ({i} --> J.i);
  for x being object st x in dom J holds J.x = ({i} --> J.i).x
  proof
    let x be object;
    assume x in dom J;
    then A2: x = i by TARSKI:def 1;
    ({i} --> J.i).i = (i .--> J.i).i by FUNCOP_1:def 9 .= J.i by FUNCOP_1:72;
    hence thesis by A2;
  end;
  hence thesis by A1, FUNCT_1:2;
end;
