reserve e for set;

theorem Th8:
  for I,J,K being set, A being ManySortedSet of I, B being
ManySortedSet of J, C being ManySortedSet of K holds A cc= B & B cc= C implies
  A cc= C
proof
  let I,J,K be set, A be ManySortedSet of I, B be ManySortedSet of J, C be
  ManySortedSet of K;
  assume that
A1: A cc= B and
A2: B cc= C;
A3: I c= J by A1;
  J c= K by A2;
  hence I c= K by A3;
  let i be set;
  assume
A4: i in I;
  then
A5: A.i c= B.i by A1;
  B.i c= C.i by A2,A3,A4;
  hence thesis by A5;
end;
