let G be Group; :: thesis:
let H, N be normal Subgroup of G; :: thesis:
assume A1: N is Subgroup of H ; :: thesis:
then consider M being strict Subgroup of G such that
A2: the carrier of M = N ` H by Th72;
for x being Element of G holds x * M c= M * x

let x be Element of G; :: thesis:
let y be object ; :: according to TARSKI:def 3 :: thesis:
assume A3: y in x * M ; :: thesis:
then reconsider y = y as Element of G ;
consider z being Element of G such that
A4: ( y = x * z & z in M ) by A3, GROUP_2:103;
z in the carrier of M by A4, STRUCT_0:def 5;
then A5: z * N c= carr H by A2, Th49;
z in z * N by GROUP_2:108;
then z in H by A5, STRUCT_0:def 5;
then (x * z) * (x ") in H by Th4;
then A6: ((x * z) * (x ")) * H = carr H by GROUP_2:113;
((x * z) * (x ")) * N c= ((x * z) * (x ")) * H by A1, GROUP_3:6;
then (x * z) * (x ") in N ` H by A6;
then A7: (x * z) * (x ") in M by A2, STRUCT_0:def 5;
((x * z) * (x ")) * x = y by A4, GROUP_3:1;
hence y in M * x by A7, GROUP_2:104; :: thesis:

end;