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 Th46:
  A1 misses A2 & A1,A2 are_weakly_separated iff A1,A2 are_separated
proof
  thus A1 misses A2 & A1,A2 are_weakly_separated implies A1,A2 are_separated
  proof
    assume that
A1: A1 misses A2 and
A2: A1,A2 are_weakly_separated;
    A1 \ A2 = A1 & A2 \ A1 = A2 by A1,XBOOLE_1:83;
    hence thesis by A2;
  end;
  assume
A3: A1,A2 are_separated;
  then A1 misses A2 by CONNSP_1:1;
  then A1 \ A2 = A1 & A2 \ A1 = A2 by XBOOLE_1:83;
  hence thesis by A3,CONNSP_1:1;
end;
