
theorem
  for L being with_suprema antisymmetric transitive RelStr for D1, D2,
  D3 being Subset of L holds (D1 "\/" D2) "\/" D3 = D1 "\/" (D2 "\/" D3)
proof
  let L be with_suprema antisymmetric transitive RelStr, D1, D2, D3 be Subset
  of L;
  thus (D1 "\/" D2) "\/" D3 c= D1 "\/" (D2 "\/" D3)
  proof
    let q be object;
    assume q in (D1 "\/" D2) "\/" D3;
    then consider a1, b1 being Element of L such that
A1: q = a1 "\/" b1 and
A2: a1 in D1 "\/" D2 and
A3: b1 in D3;
    consider a11, b11 being Element of L such that
A4: a1 = a11 "\/" b11 and
A5: a11 in D1 and
A6: b11 in D2 by A2;
    b11 "\/" b1 in D2 "\/" D3 & q = a11 "\/" (b11 "\/" b1) by A1,A3,A4,A6,
LATTICE3:14;
    hence thesis by A5;
  end;
  let q be object;
  assume q in D1 "\/" (D2 "\/" D3);
  then consider a1, b1 being Element of L such that
A7: q = a1 "\/" b1 & a1 in D1 and
A8: b1 in D2 "\/" D3;
  consider a11, b11 being Element of L such that
A9: b1 = a11 "\/" b11 & a11 in D2 and
A10: b11 in D3 by A8;
  a1 "\/" a11 in D1 "\/" D2 & q = a1 "\/" a11 "\/" b11 by A7,A9,LATTICE3:14;
  hence thesis by A10;
end;
