
theorem Th20:
  for M, N being complete LATTICE, P being Lawson correct
  TopAugmentation of [:M,N:], Q being Lawson correct TopAugmentation of M, R
  being Lawson correct TopAugmentation of N st InclPoset sigma N is continuous
  holds the TopStruct of P = [:Q,R qua TopSpace:]
proof
  let M, N be complete LATTICE, P be Lawson correct TopAugmentation of [:M,N:]
, Q be Lawson correct TopAugmentation of M, R be Lawson correct TopAugmentation
  of N such that
A1: InclPoset sigma N is continuous;
A2: the topology of P = lambda [:M,N:] by WAYBEL19:def 4
    .= the topology of [:Q,R qua TopSpace:] by A1,Th19;
A3: the RelStr of Q = the RelStr of M by YELLOW_9:def 4;
A4: the RelStr of R = the RelStr of N by YELLOW_9:def 4;
  the RelStr of P = the RelStr of [:M,N:] by YELLOW_9:def 4;
  then the carrier of P = [:the carrier of Q, the carrier of N:] by A3,
YELLOW_3:def 2
    .= the carrier of [:Q,R qua TopSpace:] by A4,BORSUK_1:def 2;
  hence thesis by A2;
end;
