theorem Th77:
  a in center G iff for b holds a * b = b * a
proof
  thus a in center G implies for b holds a * b = b * a
  proof
    assume a in center G;
    then a in the carrier of center G by STRUCT_0:def 5;
    then a in {b : for c holds b * c = c * b} by Def10;
    then ex b st a = b & for c holds b * c = c * b;
    hence thesis;
  end;
  assume for b holds a * b = b * a;
  then a in {c : for b holds c * b = b * c};
  then a in the carrier of center G by Def10;
  hence thesis by STRUCT_0:def 5;
end;
