
theorem Th19:
  for M, N being complete LATTICE, LM being Lawson correct
TopAugmentation of M, LN being Lawson correct TopAugmentation of N st InclPoset
sigma N is continuous holds the topology of [:LM,LN qua TopSpace:] = lambda [:M
  ,N:]
proof
  let M, N be complete LATTICE, LM be Lawson correct TopAugmentation of M, LN
  be Lawson correct TopAugmentation of N such that
A1: InclPoset sigma N is continuous;
  set SMN = the non empty Scott correct TopAugmentation of [:M,N:];
  set lMN = the non empty lower correct TopAugmentation of [:M,N:];
  set LMN = the non empty Lawson correct TopAugmentation of [:M,N:];
A2: [:LM,LN qua TopSpace:] = the TopStruct of LMN
  proof
    set lN = the non empty lower correct TopAugmentation of N;
    set lM = the non empty lower correct TopAugmentation of M;
    set SN = the non empty Scott correct TopAugmentation of N;
    set SM = the non empty Scott correct TopAugmentation of M;
A3: LN is Refinement of SN, lN by WAYBEL19:29;
A4: the RelStr of lN = the RelStr of N by YELLOW_9:def 4;
    (omega LMN) \/ (sigma LMN) is prebasis of LMN by WAYBEL19:def 3;
    then
A5: (omega LMN) \/ (sigma LMN) is prebasis of the TopStruct of LMN by
YELLOW12:33;
A6: the RelStr of LM = the RelStr of M by YELLOW_9:def 4;
A7: the RelStr of LN = the RelStr of N by YELLOW_9:def 4;
A8: the RelStr of LMN = the RelStr of [:M,N:] by YELLOW_9:def 4;
A9: the carrier of [:LM,LN qua TopSpace:] = [:the carrier of LM,the
    carrier of LN:] by BORSUK_1:def 2
      .= the carrier of LMN by A6,A7,A8,YELLOW_3:def 2;
A10: the topology of [:lM,lN qua TopSpace:] = omega [:M,N:] by WAYBEL19:15
      .= omega LMN by A8,WAYBEL19:3;
A11: the RelStr of SN = the RelStr of N by YELLOW_9:def 4;
    the RelStr of SM = the RelStr of M by YELLOW_9:def 4
      .= the RelStr of Sigma M by YELLOW_9:def 4;
    then
A12: the TopStruct of SM = the TopStruct of Sigma M by WAYBEL29:13;
A13: the RelStr of lM = the RelStr of M by YELLOW_9:def 4;
    the RelStr of SN = the RelStr of N by YELLOW_9:def 4
      .= the RelStr of Sigma N by YELLOW_9:def 4;
    then the TopStruct of SN = the TopStruct of Sigma N by WAYBEL29:13;
    then
A14: the topology of [:SM,SN qua TopSpace:] = the topology of [:Sigma M,(
    Sigma N) qua TopSpace:] by A12,WAYBEL29:7
      .= sigma [:M,N:] by A1,WAYBEL29:30
      .= sigma LMN by A8,YELLOW_9:52;
A15: the RelStr of SM = the RelStr of M by YELLOW_9:def 4;
A16: LM is Refinement of SM, lM by WAYBEL19:29;
    then
    [:LM,LN qua TopSpace:] is Refinement of [:SM,SN qua TopSpace:], [:lM,
    lN qua TopSpace:] by A3,A15,A11,A13,A4,YELLOW12:50;
    then the carrier of the TopStruct of LMN = (the carrier of [:SM,SN qua
    TopSpace:]) \/ (the carrier of [:lM,lN qua TopSpace:]) by A9,YELLOW_9:def 6
;
    then
    the TopStruct of LMN is Refinement of [:SM,SN qua TopSpace:], [:lM,lN
    qua TopSpace:] by A5,A14,A10,YELLOW_9:def 6;
    hence thesis by A16,A3,A15,A11,A13,A4,A9,YELLOW12:49;
  end;
  LMN is Refinement of SMN, lMN by WAYBEL19:29;
  then reconsider R = [:LM,LN qua TopSpace:] as Refinement of SMN, lMN by A2,
YELLOW12:47;
  the topology of [:LM,LN qua TopSpace:] = the topology of R;
  hence thesis by WAYBEL19:34;
end;
