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;

theorem Th39:
  for C being Subset of X holds A1,A2 are_separated implies A1 /\
  C,A2 /\ C are_separated
proof
  let C be Subset of X;
A1: A1 /\ C c= A1 & A2 /\ C c= A2 by XBOOLE_1:17;
  assume A1,A2 are_separated;
  hence thesis by A1,CONNSP_1:7;
end;
