reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X0 for non empty SubSpace of X,
  B1, B2 for Subset of X0;

theorem Th25:
  B1 = A1 & B2 = A2 implies (A1,A2 are_separated iff B1,B2 are_separated)
proof
  assume that
A1: B1 = A1 and
A2: B2 = A2;
A3: Cl B2 = (Cl A2) /\ [#]X0 by A2,PRE_TOPC:17;
A4: Cl B1 = (Cl A1) /\ [#]X0 by A1,PRE_TOPC:17;
  thus A1,A2 are_separated implies B1,B2 are_separated
  proof
    assume
A5: A1,A2 are_separated;
    then A1 misses Cl A2 by CONNSP_1:def 1;
    then A1 /\ Cl A2 = {};
    then B1 /\ Cl B2 = {} /\ [#]X0 by A1,A3,XBOOLE_1:16;
    then
A6: B1 misses Cl B2;
    Cl A1 misses A2 by A5,CONNSP_1:def 1;
    then Cl A1 /\ A2 = {};
    then Cl B1 /\ B2 = {} /\ [#]X0 by A2,A4,XBOOLE_1:16;
    then Cl B1 misses B2;
    hence thesis by A6,CONNSP_1:def 1;
  end;
  thus B1,B2 are_separated implies A1,A2 are_separated
  proof
    assume
A7: B1,B2 are_separated;
    then ((Cl A1) /\ [#]X0) misses A2 by A2,A4,CONNSP_1:def 1;
    then ((Cl A1) /\ [#]X0) /\ A2 = {};
    then
A8: (Cl A1 /\ A2) /\ [#]X0 = {} by XBOOLE_1:16;
    A1 misses ((Cl A2) /\ [#]X0) by A1,A3,A7,CONNSP_1:def 1;
    then A1 /\ ((Cl A2) /\ [#]X0) = {};
    then
A9: (A1 /\ Cl A2) /\ [#]X0 = {} by XBOOLE_1:16;
    A1 /\ Cl A2 c= A1 by XBOOLE_1:17;
    then A1 /\ Cl A2 = {} by A1,A9,XBOOLE_1:1,28;
    then
A10: A1 misses Cl A2;
    Cl A1 /\ A2 c= A2 by XBOOLE_1:17;
    then Cl A1 /\ A2 = {} by A2,A8,XBOOLE_1:1,28;
    then Cl A1 misses A2;
    hence thesis by A10,CONNSP_1:def 1;
  end;
end;
