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 Th47:
  A1 c= A2 implies A1,A2 are_weakly_separated
proof
A1: now
    assume A1 c= A2;
    then
A2: A1 \ A2 = {} by XBOOLE_1:37;
    then Cl(A1 \ A2) = {} by PRE_TOPC:22;
    then Cl(A1 \ A2) /\ (A2 \ A1) = {};
    then
A3: Cl(A1 \ A2) misses (A2 \ A1) by XBOOLE_0:def 7;
    (A1 \ A2) /\ Cl(A2 \ A1) = {} by A2;
    then (A1 \ A2) misses Cl(A2 \ A1) by XBOOLE_0:def 7;
    then A1 \ A2,A2 \ A1 are_separated by A3,CONNSP_1:def 1;
    hence thesis;
  end;
  assume A1 c= A2;
  hence thesis by A1;
end;
