reserve T for non empty TopSpace;
reserve A for Subset of T;

theorem Th57:
  for F, G being with_proper_subsets with_non-empty_elements
  Subset-Family of R^1 st F is open & G is closed holds F misses G
proof
  let F, G be with_proper_subsets with_non-empty_elements Subset-Family of R^1;
  assume
A1: F is open & G is closed;
  assume F meets G;
  then consider x being object such that
A2: x in F and
A3: x in G by XBOOLE_0:3;
  reconsider x as Subset of R^1 by A2;
  x is open & x is closed by A1,A2,A3;
  then x = {} or x = REAL by BORSUK_5:34;
  hence thesis by A2,SETFAM_1:def 12,TOPMETR:17;
end;
