theorem Th138:
  Left_Cosets (1).G = the set of all {a}
proof
  set A = the set of all {a};
  thus Left_Cosets (1).G c= A
  proof
    let x be object;
    assume
A1: x in Left_Cosets (1).G;
    then reconsider X = x as Subset of G;
    consider g such that
A2: X = g * (1).G by A1,Def15;
    x = {g} by A2,Th110;
    hence thesis;
  end;
  let x be object;
  assume x in A;
  then consider a such that
A3: x = {a};
  a * (1).G = {a} by Th110;
  hence thesis by A3,Def15;
end;
