reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem Th9:
  for G being Group, N,N1,N2 being Subgroup of G st
  the carrier of N = N1 * N2 holds N1 is Subgroup of N & N2 is Subgroup of N
proof
  let G be Group, N,N1,N2 be Subgroup of G;
  assume
A1:the carrier of N = N1 * N2;
  for x be Element of G st x in N1 holds x in N
  proof
    let x be Element of G;
    assume
A2: x in N1;
A3: 1_ G in N2 by GROUP_2:46;
    x = x * 1_ G by GROUP_1:def 4;
    then x in N1 * N2 by A2,A3,Th6;
    hence thesis by A1,STRUCT_0:def 5;
  end;
  hence N1 is Subgroup of N  by GROUP_2:58;
  for y be Element of G st y in N2 holds y in N
  proof
    let y be Element of G;
    assume
A4: y in N2;
A5: 1_ G in N1 by GROUP_2:46;
    y = 1_ G * y by GROUP_1:def 4;
    then y in N1 * N2 by A4,A5,Th6;
    hence thesis by A1,STRUCT_0:def 5;
  end;
  hence N2 is Subgroup of N  by GROUP_2:58;
end;
