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;

theorem Th34:
  A1 is closed & A2 is closed implies (A1 misses A2 iff A1,A2 are_separated)
proof
  assume
A1: A1 is closed & A2 is closed;
  thus A1 misses A2 implies A1,A2 are_separated
  proof
    assume
A2: A1 misses A2;
    Cl A1 = A1 & Cl A2 = A2 by A1,PRE_TOPC:22;
    hence thesis by A2,CONNSP_1:def 1;
  end;
  thus thesis by CONNSP_1:1;
end;
