 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;

theorem
  G is unital iff for a holds
    (the_unity_wrt the multF of G)*a = a &
    a*(the_unity_wrt the multF of G) = a
proof
  set u = the_unity_wrt the multF of G;
  thus G is unital implies for b holds u*b = b & b*u = b
  proof
    given a such that
A1: a is_a_unity_wrt op(G);
    let b;
    u = a by A1,BINOP_1:def 8;
    hence thesis by A1,BINOP_1:3;
  end;
  assume
A2: u*b = b & b*u = b;
  take a = u;
  thus a is_a_left_unity_wrt the multF of G
  proof
    let b;
    a*b = b by A2;
    hence thesis;
  end;
  let b;
  b*a = b by A2;
  hence thesis;
end;
