theorem Th10:
  (add_inverse G)" A = -A
proof
  set f = add_inverse G;
A1: dom f = the carrier of G by FUNCT_2:def 1;
  hereby
    let x be object;
    assume
A2: x in f"A;
    then reconsider g = x as Element of G;
    f.x in A by A2,FUNCT_1:def 7;
    then -(f.g) in -A;
    then - -g in -A by Def6;
    hence x in -A;
  end;
  let x be object;
  assume x in -A;
  then consider g being Element of G such that
A3: x = -g & g in A;
  f.(-g) = - -g by Def6
    .= g;
  hence thesis by A1,A3,FUNCT_1:def 7;
end;
