
theorem
  for L being antisymmetric with_suprema RelStr, A being upper Subset of
  L for B, C being Subset of L holds (A \/ B) "\/" (A \/ C) c= A \/ (B "\/" C)
proof
  let L be antisymmetric with_suprema RelStr, A be upper Subset of L, B, C be
  Subset of L;
  let q be object;
  assume q in (A \/ B) "\/" (A \/ C);
  then consider x, y being Element of L such that
A1: q = x "\/" y and
A2: x in A \/ B & y in A \/ C;
A3: x <= x "\/" y by YELLOW_0:22;
A4: y <= x "\/" y by YELLOW_0:22;
  per cases by A2,XBOOLE_0:def 3;
  suppose
    x in A & y in A;
    then q in A by A1,A3,WAYBEL_0:def 20;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    x in A & y in C;
    then q in A by A1,A3,WAYBEL_0:def 20;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    x in B & y in A;
    then q in A by A1,A4,WAYBEL_0:def 20;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    x in B & y in C;
    then x "\/" y in B "\/" C;
    hence thesis by A1,XBOOLE_0:def 3;
  end;
end;
