reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
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 addGroup;
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
  Right_Cosets (0).G = the set of all {a}
proof
  set A = the set of all {a} ;
  thus Right_Cosets (0).G c= A
  proof
    let x be object;
    assume
A1: x in Right_Cosets (0).G;
    then reconsider X = x as Subset of G;
    consider g such that
A2: X = (0).G + g by A1,Def16;
    x = {g} by A2,Th110;
    hence thesis;
  end;
  let x be object;
  assume x in A;
  then consider a such that
A3: x = {a};
  (0).G + a = {a} by Th110;
  hence thesis by A3,Def16;
end;
