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;

theorem
  addMagma (# REAL, addreal #) is Abelian addGroup
proof
  reconsider G = addMagma (# REAL, addreal #) as addGroup by Th3;
  G is Abelian
  proof
    let h,g be Element of G;
    reconsider A = h, B = g as Real;
    thus h + g = B + A by BINOP_2:def 9
      .= g + h by BINOP_2:def 9;
  end;
  hence thesis;
end;
