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;
reserve N1,N2 for Subgroup of G;

theorem Th71:
  for H being Subgroup of G, N being normal Subgroup of G
  ex M being strict Subgroup of G st the carrier of M = N ~ H
proof
  let H be Subgroup of G, N be normal Subgroup of G;
A1:1_G in N ~ H
  proof
    1_G in H by GROUP_2:46; then
A2: 1_G in carr(H) by STRUCT_0:def 5;
    1_G in 1_G * N by GROUP_2:108;
    then 1_G * N meets carr(H) by A2,XBOOLE_0:3;
    hence thesis;
  end;
A3:for x,y being Element of G holds x in N ~ H & y in N ~ H
   implies x * y in N ~ H
   proof
     let x,y be Element of G;
     assume that
A4:  x in N ~ H and
A5:  y in N ~ H;
     consider a be Element of G such that
A6:  a in x * N & a in H by A4,Th51,Th3;
     consider b be Element of G such that
A7:  b in y * N & b in H by A5,Th51,Th3;
     (x * N) * (y * N) = (x * y) * N & (a * N) * (b * N) = (a * b) * N by Th1;
     then
A8: a * b in (x * y) * N by A6,A7;
     a * b in H by A6,A7,GROUP_2:50;
     then a * b in carr(H) by STRUCT_0:def 5;
     then (x * y) * N meets carr(H) by A8,XBOOLE_0:3;
     hence thesis;
   end;
for x being Element of G holds x in N ~ H implies x" in N ~ H
  proof
    let x be Element of G;
    assume x in N ~ H;
    then consider a be Element of G such that
A9:a in x * N & a in H by Th3,Th51;
    consider a1 be Element of G such that
A10:a = x * a1 & a1 in N by A9,GROUP_2:103;
A11:a1" in N by A10,GROUP_2:51;
    a" = a1" * x" by A10,GROUP_1:17;
    then a" in N * x" by A11,GROUP_2:104; then
A12:a" in x" * N by GROUP_3:117;
    a" in H by A9,GROUP_2:51;
    then a" in carr(H) by STRUCT_0:def 5;
    then x" * N meets carr(H) by A12,XBOOLE_0:3;
    hence thesis;
  end;
  hence thesis by A1,A3,GROUP_2:52;
end;
