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 Th24:
  for A1, A2, C1, C2 being Subset of X st C1 c= A1 & C2 c= A2 & C1
  /\ C2 = A1 /\ A2 holds A1,A2 are_weakly_separated implies C1,C2
  are_weakly_separated
proof
  let A1, A2, C1, C2 be Subset of X;
  assume C1 c= A1 & C2 c= A2;
  then
A1: C1 \ (C1 /\ C2) c= A1 \ (C1 /\ C2) & C2 \ (C1 /\ C2) c= A2 \ (C1 /\ C2)
  by XBOOLE_1:33;
  assume
A2: C1 /\ C2 = A1 /\ A2;
  assume A1,A2 are_weakly_separated;
  then A1 \ (C1 /\ C2),A2 \ (C1 /\ C2) are_separated by A2,Th23;
  then C1 \ (C1 /\ C2),C2 \ (C1 /\ C2) are_separated by A1,CONNSP_1:7;
  hence thesis by Th23;
end;
