
theorem
  for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
  for X1 being Subset of L1, X2 being Subset of L2 st X1 = X2 holds
  (X1 is lower implies X2 is lower) & (X1 is upper implies X2 is upper)
proof
  let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
  let X1 be Subset of L1, X2 be Subset of L2;
  assume
A2: X1 = X2;
  hereby
    assume
A3: X1 is lower;
A4: downarrow X1 = downarrow X2 by A1,A2,Th13;
    downarrow X1 c= X1 by A3,Th23;
    hence X2 is lower by A2,A4,Th23;
  end;
  assume
A5: X1 is upper;
A6: uparrow X1 = uparrow X2 by A1,A2,Th13;
  uparrow X1 c= X1 by A5,Th24;
  hence thesis by A2,A6,Th24;
end;
