
theorem Th8: :: 1.2.  LEMMA, p. 179
:: LowerAdj preserves contravariantly composition
  for L1,L2,L3 being complete LATTICE
  for g1 being infs-preserving Function of L1,L2
  for g2 being infs-preserving Function of L2,L3 holds
  LowerAdj (g2*g1) = (LowerAdj g1)*(LowerAdj g2)
proof
  let L1,L2,L3 be complete LATTICE;
  let g1 be infs-preserving Function of L1,L2;
  let g2 be infs-preserving Function of L2,L3;
A1: [g1, LowerAdj g1] is Galois by Def1;
  [g2, LowerAdj g2] is Galois by Def1;
  then
A2: [g2*g1, (LowerAdj g1)*(LowerAdj g2)] is Galois by A1,WAYBEL15:5;
  g2*g1 is infs-preserving by WAYBEL20:25;
  hence thesis by A2,Def1;
end;
