reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

theorem
  for N being normal Subgroup of G holds Left_Cosets N = Right_Cosets N
proof
  let N be normal Subgroup of G;
  thus Left_Cosets N c= Right_Cosets N
  proof
    let x be object;
    assume x in Left_Cosets N;
    then consider a such that
A1: x = a * N by GROUP_2:def 15;
    x = N * a by A1,Th117;
    hence thesis by GROUP_2:def 16;
  end;
  let x be object;
  assume x in Right_Cosets N;
  then consider a such that
A2: x = N * a by GROUP_2:def 16;
  x = a * N by A2,Th117;
  hence thesis by GROUP_2:def 15;
end;
