
theorem Th9: :: 1.2.  LEMMA, p. 179
:: UpperAdj preserves contravariantly composition
  for L1,L2,L3 being complete LATTICE
  for d1 being sups-preserving Function of L1,L2
  for d2 being sups-preserving Function of L2,L3 holds
  UpperAdj (d2*d1) = (UpperAdj d1)*(UpperAdj d2)
proof
  let L1,L2,L3 be complete LATTICE;
  let d1 be sups-preserving Function of L1,L2;
  let d2 be sups-preserving Function of L2,L3;
A1: [UpperAdj d1, d1] is Galois by Def2;
  [UpperAdj d2, d2] is Galois by Def2;
  then
A2: [(UpperAdj d1)*(UpperAdj d2), d2*d1] is Galois by A1,WAYBEL15:5;
  d2*d1 is sups-preserving by WAYBEL20:27;
  hence thesis by A2,Def2;
end;
