theorem Th48:
  h = g implies -h = -g
proof
  reconsider g9 = -h as Element of G by Th41,STRUCT_0:def 5;
A1: h + -h = 0_H by Def5;
  assume h = g;
  then g + g9 = 0_H by A1,Th43
    .= 0_G by Th44;
  hence thesis by Th11;
end;
