theorem Th15:
  h*.:P = h * P
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 = h * x by Def1;
    hence y in h * P by A2,GROUP_2:27;
  end;
  let y be object;
  assume y in h * P;
  then consider s being Element of H such that
A3: y = h * s & s in P by GROUP_2:27;
  dom f = the carrier of H & f.s = h * s by Def1,FUNCT_2:def 1;
  hence thesis by A3,FUNCT_1:def 6;
end;
