reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;

theorem
  for R, S, T being TopStruct holds R is Refinement of S, T iff the
  TopStruct of R is Refinement of S, T
proof
  let R, S, T be TopStruct;
  hereby
    assume
A1: R is Refinement of S, T;
    then reconsider R1 = R as TopSpace;
    (the topology of S) \/ (the topology of T) is prebasis of R by A1,
YELLOW_9:def 6;
    then
A2: (the topology of S) \/ (the topology of T) is prebasis of the TopStruct
    of R1 by Th33;
    the carrier of the TopStruct of R1 = (the carrier of S) \/ (the
    carrier of T) by A1,YELLOW_9:def 6;
    hence the TopStruct of R is Refinement of S, T by A2,YELLOW_9:def 6;
  end;
  assume
A3: the TopStruct of R is Refinement of S, T;
  then reconsider R1 = R as TopSpace by TEX_2:7;
  (the topology of S) \/ (the topology of T) is prebasis of the TopStruct
  of R by A3,YELLOW_9:def 6;
  then
A4: (the topology of S) \/ (the topology of T) is prebasis of R1 by Th33;
  the carrier of R1 = (the carrier of S) \/ (the carrier of T) by A3,
YELLOW_9:def 6;
  hence thesis by A4,YELLOW_9:def 6;
end;
