reserve A, B, X, Y for set;

theorem
  for L1, L2 being non empty RelStr, A being SubRelStr of L1, J being
SubRelStr of L2 st the RelStr of L1 = the RelStr of L2 & the RelStr of A = the
  RelStr of J & A is join-inheriting holds J is join-inheriting
proof
  let L1, L2 be non empty RelStr, A be SubRelStr of L1, J be SubRelStr of L2
  such that
A1: the RelStr of L1 = the RelStr of L2 and
A2: the RelStr of A = the RelStr of J and
A3: for x, y being Element of L1 st x in the carrier of A & y in the
  carrier of A & ex_sup_of {x,y},L1 holds sup {x,y} in the carrier of A;
  let x, y be Element of L2 such that
A4: x in the carrier of J & y in the carrier of J and
A5: ex_sup_of {x,y},L2;
  reconsider x1 = x, y1 = y as Element of L1 by A1;
  sup {x1,y1} in the carrier of A by A1,A2,A3,A4,A5,YELLOW_0:14;
  hence thesis by A1,A2,A5,YELLOW_0:26;
end;
