theorem Th54:
  (for y being Element of Y holds g.y = F.(x,f.y)) implies g = F
  [;](x,f)
proof
  assume
A1: for y being Element of Y holds g.y = F.(x,f.y);
  now
    let y be Element of Y;
    thus g.y = F.(x,f.y) by A1
      .= (F[;](x,f)).y by Th53;
  end;
  hence thesis by FUNCT_2:63;
end;
