
theorem Th1:
  for L being complete LATTICE, X,Y being Subset of L st Y is_coarser_than X
  holds "/\"(X,L) <= "/\"(Y,L)
proof
  let L be complete LATTICE, X,Y be Subset of L such that
A1: Y is_coarser_than X;
  "/\"(X,L) is_<=_than Y
  proof
    let y be Element of L;
    assume y in Y;
    then consider x being Element of L such that
A2: x in X and
A3: x <= y by A1;
    "/\"(X,L) <= x by A2,YELLOW_2:22;
    hence "/\"(X,L) <= y by A3,YELLOW_0:def 2;
  end;
  hence thesis by YELLOW_0:33;
end;
