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 Th41:
  for B being Subset of X holds A1,B are_separated & A2,B
  are_separated iff A1 \/ A2,B are_separated
proof
  let B be Subset of X;
  A1 \/ A2,B are_separated implies A1,B are_separated & A2,B are_separated
  proof
A1: A1 c= A1 \/ A2 & A2 c= A1 \/ A2 by XBOOLE_1:7;
    assume A1 \/ A2,B are_separated;
    hence thesis by A1,CONNSP_1:7;
  end;
  hence
  A1,B are_separated & A2,B are_separated iff A1 \/ A2,B are_separated by
CONNSP_1:8;
end;
