theorem Th113:
  [:X1,X2:] c= [:Y1,Y2:] & [:X1,X2:] <> {} implies X1 c= Y1 & X2 c= Y2
proof
  assume that
A1: [:X1,X2:] c= [:Y1,Y2:] and
A2: [:X1,X2:] <> {};
A3: [:X1,X2:] = [:X1,X2:] /\ [:Y1,Y2:] by A1,XBOOLE_1:28
    .= [:X1/\Y1,X2/\Y2:] by Th99;
  X1 <> {} & X2 <> {} by A2,Th89;
  then X1 = X1/\Y1 & X2 = X2/\Y2 by A3,Th109;
  hence thesis by XBOOLE_1:17;
end;
