
theorem
  for L being antisymmetric reflexive with_suprema RelStr for A, B, C
  being Subset of L holds A \/ (B "\/" C) c= (A \/ B) "\/" (A \/ C)
proof
  let L be antisymmetric reflexive with_suprema RelStr, A, B, C be Subset of L;
  let q be object such that
A1: q in A \/ (B "\/" C);
  per cases by A1,XBOOLE_0:def 3;
  suppose
A2: q in A;
    then reconsider q1 = q as Element of L;
    q1 <= q1;
    then
A3: q1 = q1 "\/" q1 by YELLOW_0:24;
    q1 in A \/ B & q1 in A \/ C by A2,XBOOLE_0:def 3;
    hence thesis by A3;
  end;
  suppose
    q in B "\/" C;
    then consider b, c being Element of L such that
A4: q = b "\/" c and
A5: b in B & c in C;
    b in A \/ B & c in A \/ C by A5,XBOOLE_0:def 3;
    hence thesis by A4;
  end;
end;
