
theorem Th32:
  for L being complete LATTICE holds (for S being complete LATTICE
  holds sigma [:S, L:] = the topology of [:Sigma S, Sigma L:]) iff for S being
  complete LATTICE holds Sigma [:S, L:] = Omega [:Sigma S, Sigma L:]
proof
  let L be complete LATTICE;
  hereby
    assume
A1: for S being complete LATTICE holds sigma [:S, L:] = the topology
    of [:Sigma S, Sigma L:];
    let S be complete LATTICE;
    the TopStruct of Sigma [:S, L:] = [:Sigma S, Sigma L:] by A1,Th31;
    then Omega Sigma [:S, L:] = Omega [:Sigma S, Sigma L:] by WAYBEL25:13;
    hence Sigma [:S, L:] = Omega [:Sigma S, Sigma L:] by WAYBEL25:15;
  end;
  assume
A2: for S being complete LATTICE holds Sigma [:S, L:] = Omega [:Sigma S,
  Sigma L:];
  let S be complete LATTICE;
  Sigma [:S, L:] = Omega [:Sigma S, Sigma L:] by A2;
  then the TopStruct of Sigma [:S, L:] = [:Sigma S, Sigma L:] by WAYBEL25:def 2
;
  hence thesis by YELLOW_9:51;
end;
