 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for Group-like non empty multMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for Group;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;

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;
