reserve A, B, X, Y for set;

theorem Th10:
  for L1 being sup-Semilattice, L2 being non empty RelStr for x, y
  being Element of L1, x1, y1 being Element of L2 st the RelStr of L1 = the
  RelStr of L2 & x = x1 & y = y1 holds x "\/" y = x1 "\/" y1
proof
  let L1 be sup-Semilattice, L2 be non empty RelStr, x, y be Element of L1, x1
  , y1 be Element of L2 such that
A1: the RelStr of L1 = the RelStr of L2 and
A2: x = x1 & y = y1;
A3: L2 is with_suprema Poset by A1,WAYBEL_8:12,13,14,YELLOW_7:15;
A4: ex_sup_of {x,y}, L1 by YELLOW_0:20;
  thus x "\/" y = sup{x,y} by YELLOW_0:41
    .= sup{x1,y1} by A1,A2,A4,YELLOW_0:26
    .= x1 "\/" y1 by A3,YELLOW_0:41;
end;
