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;

theorem Th22:
  for A1, A2, C1, C2 being Subset of X st C1 c= A1 & C2 c= A2 & C1
  \/ C2 = A1 \/ A2 holds C1,C2 are_weakly_separated implies A1,A2
  are_weakly_separated
proof
  let A1, A2, C1, C2 be Subset of X;
  assume C1 c= A1 & C2 c= A2;
  then
A1: (A1 \/ A2) \ A1 c= (A1 \/ A2) \ C1 & (A1 \/ A2) \ A2 c= (A1 \/ A2) \ C2
  by XBOOLE_1:34;
  assume
A2: C1 \/ C2 = A1 \/ A2;
  assume C1,C2 are_weakly_separated;
  then (A1 \/ A2) \ C1,(A1 \/ A2) \ C2 are_separated by A2,Th21;
  then (A1 \/ A2) \ A1,(A1 \/ A2) \ A2 are_separated by A1,CONNSP_1:7;
  hence thesis by Th21;
end;
