theorem Th16:
  (*h).:P = P * h
proof
  set f = *h;
  hereby
    let y be object;
    assume y in f.:P;
    then consider x being object such that
A1: x in dom f and
A2: x in P & y = f.x by FUNCT_1:def 6;
    reconsider x as Element of H by A1;
    f.x = x * h by Def2;
    hence y in P * h by A2,GROUP_2:28;
  end;
  let y be object;
  assume y in P * h;
  then consider s being Element of H such that
A3: y = s * h & s in P by GROUP_2:28;
  dom f = the carrier of H & f.s = s * h by Def2,FUNCT_2:def 1;
  hence thesis by A3,FUNCT_1:def 6;
end;
