theorem Th8:
  g <> {} & g - c1 = g - c2 implies c1 = c2
proof
  assume that
A1: g <> {} and
A2: g-c1 = g-c2;
  consider x being object such that
A3: x in dom g by A1,XBOOLE_0:def 1;
  dom g = dom(g-c2) by VALUED_1:def 2;
  then
A4: (g-c2).x = g.x-c2 by A3,VALUED_1:def 2;
  dom g = dom(g-c1) by VALUED_1:def 2;
  then (g-c1).x = g.x-c1 by A3,VALUED_1:def 2;
  hence c1 = c2 by A2,A4;
end;
