
theorem
  for I be non empty set, M1, M2 be non-empty ManySortedSet of I holds
  M1|support M1 = M2|support M2 implies M1 = M2
proof
  let I be non empty set, M1, M2 be non-empty ManySortedSet of I;
A1: dom M1 = I by PARTFUN1:def 2;
A2: dom M2 = I by PARTFUN1:def 2;
  assume
A3: M1|support M1 = M2|support M2;
  for x be object st x in I holds M1.x = M2.x
  proof
    let x be object;
A4: dom (M2|support M2) = dom M2 /\ support M2 by RELAT_1:61;
    assume
A5: x in I;
    then M1.x is non empty;
    then
A6: x in support M1 by A5;
    M2.x is non empty by A5;
    then x in support M2 by A5;
    then
A7: x in dom (M2|support M2) by A2,A5,A4,XBOOLE_0:def 4;
    dom (M1|support M1) = dom M1 /\ support M1 by RELAT_1:61;
    then x in dom (M1|support M1) by A1,A5,A6,XBOOLE_0:def 4;
    then M1.x = (M2|support M2).x by A3,FUNCT_1:47
      .= M2.x by A7,FUNCT_1:47;
    hence thesis;
  end;
  hence thesis;
end;
