 reserve m,n for Nat;
 reserve i,j for Integer;
 reserve S for non empty multMagma;
 reserve r,r1,r2,s,s1,s2,t for Element of S;

theorem Th3:
  multMagma (# REAL, addreal #) is associative Group-like
proof
  set G = multMagma (# REAL, addreal #);
  thus for h,g,f being Element of G holds h * g * f = h * (g * f)
  proof
    let h,g,f be Element of G;
    reconsider A = h, B = g, C = f as Real;
A1: g * f = B + C by BINOP_2:def 9;
    h * g = A + B by BINOP_2:def 9;
    hence h * g * f = A + B + C by BINOP_2:def 9
      .= A + (B + C)
      .= h * (g * f) by A1,BINOP_2:def 9;
  end;
  reconsider e = 0 as Element of G by XREAL_0:def 1;
  take e;
  let h be Element of G;
  reconsider A = h as Real;
  thus h * e = A + 0 by BINOP_2:def 9
    .= h;
  thus e * h = 0 + A by BINOP_2:def 9
    .= h;
  reconsider g = - A as Element of G by XREAL_0:def 1;
  take g;
  thus h * g = A + (- A) by BINOP_2:def 9
    .= e;
  thus g * h = (- A) + A by BINOP_2:def 9
    .= e;
end;
