reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;

theorem Th48:
  for A1,A2 being Subset of X holds A1 is closed & A2 is closed
  implies A1,A2 are_weakly_separated
proof
  let A1,A2 be Subset of X;
  assume that
A1: A1 is closed and
A2: A2 is closed;
  Cl(A2 \ A1) c= A2 by A2,TOPS_1:5,XBOOLE_1:36;
  then Cl(A2 \ A1) \ A2 = {} by XBOOLE_1:37;
  then
A3: (A1 \ A2) misses Cl(A2 \ A1) by Lm1;
  Cl(A1 \ A2) c= A1 by A1,TOPS_1:5,XBOOLE_1:36;
  then Cl(A1 \ A2) \ A1 = {} by XBOOLE_1:37;
  then Cl(A1 \ A2) misses (A2 \ A1) by Lm1;
  then A1 \ A2,A2 \ A1 are_separated by A3,CONNSP_1:def 1;
  hence thesis;
end;
