
theorem Th31:
  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 the TopStruct of Sigma [:S, L:] = [: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:];
    a4112[L]
    proof
      let SL be Scott TopAugmentation of L;
      let S be complete LATTICE;
      let 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 thesis by A1;
    end;
    hence for S being complete LATTICE holds the TopStruct of Sigma [:S, L:] =
    [:Sigma S, Sigma L:] by Lm9;
  end;
  assume
A3: for S being complete LATTICE holds the TopStruct of Sigma [:S, L:] =
  [:Sigma S, Sigma L:];
  let S be complete LATTICE;
  the TopStruct of Sigma [:S, L:] = [:Sigma S, Sigma L:] by A3;
  hence thesis by YELLOW_9:51;
end;
