
theorem Th30:
  for L being complete LATTICE holds InclPoset sigma L is
continuous iff for S being complete LATTICE holds sigma [:S, L:] = the topology
  of [:Sigma S, Sigma L:]
proof
  let L be complete LATTICE;
  thus InclPoset sigma L is continuous implies for S being complete LATTICE
  holds sigma [:S, L:] = the topology of [:Sigma S, Sigma L:] by Lm10;
  assume
A1: for S being complete LATTICE holds sigma [:S, L:] = the topology of
  [:Sigma S, Sigma L:];
  now
    let SL be Scott TopAugmentation of L;
    let S be complete LATTICE, SS be Scott TopAugmentation of S;
    the RelStr of SL = the RelStr of L & the RelStr of Sigma L = the
    RelStr of L by YELLOW_9:def 4;
    then
A2: the TopStruct of Sigma L = the TopStruct of SL by Th13;
    the RelStr of SS = the RelStr of S & the RelStr of Sigma S = the
    RelStr of S by YELLOW_9:def 4;
    then the TopStruct of Sigma S = the TopStruct of SS by Th13;
    then [:SS, SL:] = [:Sigma S, Sigma L:] by A2,Th7;
    hence sigma [:S,L:] = the topology of [:SS,SL:] by A1;
  end;
  hence thesis by Lm11;
end;
