reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;

theorem Th4:
  for L being join-absorbing join-commutative join-associative non
empty LattStr, a, b, c, d being Element of L st a [= b & c [= d holds a "\/" c
  [= b "\/" d
proof
  let L be join-absorbing join-commutative join-associative non empty LattStr
  , a, b, c, d be Element of L;
  assume a [= b;
  then
A1: b = a "\/" b;
  assume c [= d;
  then b "\/" d = (a "\/" b) "\/" (c "\/" d) by A1
    .= ((b "\/" a) "\/" c) "\/" d by LATTICES:def 5
    .= (b "\/" (a "\/" c)) "\/" d by LATTICES:def 5
    .= (a "\/" c) "\/" (b "\/" d) by LATTICES:def 5;
  hence thesis;
end;
