
theorem Th4:
  for I being non empty set, X being disjoint_valued ManySortedSet
of I, D being non-empty ManySortedSet of I for F1,F2 be ManySortedFunction of X
  ,D st Flatten F1 = Flatten F2 holds F1 = F2
proof
  let I be non empty set, X be disjoint_valued ManySortedSet of I, D be
  non-empty ManySortedSet of I;
  let F1,F2 be ManySortedFunction of X,D;
  assume
A1: Flatten F1 = Flatten F2;
  now
    let i be object;
    assume
A2: i in I;
    then reconsider Di=D.i as non empty set;
    reconsider f1 = F1.i, f2 = F2.i as Function of X.i,Di by A2,PBOOLE:def 15;
    now
      let x be object;
      assume
A3:   x in X.i;
      hence f1.x = (Flatten F1).x by A2,Def1
        .= f2.x by A1,A2,A3,Def1;
    end;
    hence F1.i = F2.i by FUNCT_2:12;
  end;
  hence thesis;
end;
