theorem Th9:
  (add_inverse G).:A = -A
proof
  set f = add_inverse G;
  hereby
    let y be object;
    assume y in f.:A;
    then consider x being object such that
A1: x in the carrier of G and
A2: x in A and
A3: y = f.x by FUNCT_2:64;
    reconsider x as Element of G by A1;
    y = -x by A3,Def6;
    hence y in -A by A2;
  end;
  let y be object;
  assume y in -A;
  then consider g being Element of G such that
A4: y = -g & g in A;
  f.g = -g by Def6;
  hence thesis by A4,FUNCT_2:35;
end;
