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 Th7:
  for G being Group, N1,N2 being strict normal Subgroup of G
   ex M being strict Subgroup of G st the carrier of M = N1 * N2
proof
  let G be Group, N1,N2 be strict normal Subgroup of G;
A1:1_G in N1 * N2
  proof
A2: 1_G in N1 & 1_G in N2 by GROUP_2:46;
    1_G * 1_G = 1_G by GROUP_1:def 4;
    hence thesis by A2,Th6;
  end;
A3:for x,y being Element of G holds x in N1 * N2 & y in N1 * N2
   implies x * y in N1 * N2
   proof
     let x,y be Element of G;
     assume that
A4:  x in N1 * N2 and
A5:  y in N1 * N2;
     consider a,b be Element of G such that
A6:  x = a * b & a in N1 & b in N2 by A4,Th6;
     consider c,d be Element of G such that
A7:  y = c * d & c in N1 & d in N2 by A5,Th6;
A8:  x * y = ((a * b) * c) * d by A6,A7,GROUP_1:def 3
          .= a * (b * c) * d by GROUP_1:def 3;
     b * c in N2 * N1 by A6,A7,Th6;
     then b * c in N1 * N2 by GROUP_3:125;
     then consider b9,c9 be Element of G such that
A9:  b * c = b9 * c9 & b9 in N1 & c9 in N2 by Th6;
A10: x * y = ((a * b9) * c9) * d by A8,A9,GROUP_1:def 3
          .= (a * b9) * (c9 * d) by GROUP_1:def 3;
A11: a * b9 in N1 by A6,A9,GROUP_2:50;
     c9* d in N2 by A7,A9,GROUP_2:50;
     hence thesis by A10,A11,Th6;
   end;
for x being Element of G holds x in N1 * N2 implies x" in N1 * N2
  proof
    let x be Element of G;
    assume x in N1 * N2;
    then consider a,b be Element of G such that
A12: x = a * b & a in N1 & b in N2 by Th6;
A13: x" = b" * a" by A12,GROUP_1:17;
    b" in N2 & a" in N1 by A12,GROUP_2:51;
     then
    x" in N2 * N1 by A13,Th6;
    hence thesis by GROUP_3:125;
  end;
  hence thesis by A1,A3,GROUP_2:52;
end;
