reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;

theorem Th7:
  K,L are_separated & K1 c= K & L1 c= L implies K1,L1 are_separated
proof
  assume that
A1: (Cl K) /\ L = {} and
A2: K /\ Cl L = {} and
A3: K1 c= K and
A4: L1 c= L;
  Cl L1 c= Cl L by A4,PRE_TOPC:19;
  then K1 /\ Cl L1 = {}TS by A2,A3,XBOOLE_1:3,27;
  then
A5: K1 misses Cl L1;
  Cl K1 c= Cl K by A3,PRE_TOPC:19;
  then (Cl K1) /\ L1 = {}TS by A1,A4,XBOOLE_1:3,27;
  then (Cl K1) misses L1;
  hence thesis by A5;
end;
