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

theorem
  X1,X2 are_separated iff ex Y1, Y2 being non empty SubSpace of X st Y1,
  Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & (Y1
  misses Y2 or Y1 meet Y2 misses X1 union X2)
proof
  thus X1,X2 are_separated implies ex Y1, Y2 being non empty SubSpace of X st
Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & (Y1
  misses Y2 or Y1 meet Y2 misses X1 union X2)
  proof
    assume X1,X2 are_separated;
    then consider Y1, Y2 being open non empty SubSpace of X such that
A1: X1 is SubSpace of Y1 & X2 is SubSpace of Y2 &( Y1 misses Y2 or Y1
    meet Y2 misses X1 union X2) by Th77;
    take Y1,Y2;
    thus thesis by A1,Th81;
  end;
  given Y1, Y2 being non empty SubSpace of X such that
A2: Y1,Y2 are_weakly_separated and
A3: X1 is SubSpace of Y1 & X2 is SubSpace of Y2 and
A4: Y1 misses Y2 or Y1 meet Y2 misses X1 union X2;
  reconsider C2 = the carrier of Y2 as Subset of X by Th1;
  reconsider C1 = the carrier of Y1 as Subset of X by Th1;
  now
    let A1, A2 be Subset of X such that
A5: A1 = the carrier of X1 & A2 = the carrier of X2;
    now
      per cases;
      suppose
        Y1 misses Y2;
        then Y1,Y2 are_separated by A2,Th78;
        then
A6:     C1,C2 are_separated;
        A1 c= C1 & A2 c= C2 by A3,A5,Th4;
        hence A1,A2 are_separated by A6,CONNSP_1:7;
      end;
      suppose
A7:     not Y1 misses Y2;
        ex B1, B2 being Subset of X st B1,B2 are_weakly_separated & A1 c=
        B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2
        proof
          take C1,C2;
          the carrier of Y1 meet Y2 = C1 /\ C2 & the carrier of X1 union
          X2 = A1 \/ A2 by A5,A7,Def2,Def4;
          hence thesis by A2,A3,A4,A5,A7,Th4;
        end;
        hence A1,A2 are_separated by Th62;
      end;
    end;
    hence A1,A2 are_separated;
  end;
  hence thesis;
end;
