
theorem Th66: :: PROPOSITION 4.3.(iv)
  for L be lower-bounded continuous sup-Semilattice for B be
  with_bottom CLbasis of L holds supMap (subrelstr B) is infs-preserving
  sups-preserving
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
A1: SupMap L is sups-preserving by WAYBEL13:33;
  thus supMap (subrelstr B) is infs-preserving by Th64,WAYBEL_1:12;
A2: supMap (subrelstr B) = (SupMap L)*(idsMap (subrelstr B)) by Th62;
  idsMap (subrelstr B) is sups-preserving by Th61;
  hence thesis by A2,A1,WAYBEL20:27;
end;
