reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;

theorem Th13:
  for x being object holds
  for N being normal Subgroup of G holds x in Cosets N implies
  ex a st x = a * N & x = N * a
proof let x be object;
  let N be normal Subgroup of G;
  assume x in Cosets N;
  then consider a such that
A1: x = a * N by GROUP_2:def 15;
  x = N * a by A1,GROUP_3:117;
  hence thesis by A1;
end;
