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 Th1:
  for L being join-associative join-commutative meet-commutative
  join-absorbing meet-absorbing non empty LattStr, p, q, r being Element of L
  st p [= q holds r "\/" p [= r "\/" q
proof
  let L be join-associative join-commutative meet-commutative join-absorbing
  meet-absorbing non empty LattStr, p, q, r be Element of L;
  assume
A1: p "\/" q = q;
  thus (r "\/" p) "\/" (r "\/" q) = r "\/" (r "\/" p) "\/" q by LATTICES:def 5
    .= r "\/" r "\/" p "\/" q by LATTICES:def 5
    .= r "\/" p "\/" q
    .= r "\/" q by A1,LATTICES:def 5;
end;
