reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;

theorem Th13:
  X c= Y & Y c= Z implies X c= Z
proof
  assume that
A1: X c= Y and
A2: Y c= Z;
  let i be object such that
A3: i in I;
  let e be object;
  assume
A4: e in X.i;
  X.i c= Y.i by A1,A3;
  then
A5: e in Y.i by A4;
  Y.i c= Z.i by A2,A3;
  hence thesis by A5;
end;
