theorem Th37:
  g * h = h * g implies (g * h) |^ i = g |^ i * (h |^ i)
proof
  assume
A1: g * h = h * g;
  per cases;
  suppose
A2: i >= 0;
    then
A3: h |^ i = h |^ |.i.| by Def8;
    (g * h) |^ i = (g * h) |^ |.i.| & g |^ i = g |^ |.i.| by A2,Def8;
    hence thesis by A1,A3,Lm11;
  end;
  suppose
A4: i < 0;
    hence (g * h) |^ i = ((h * g) |^ |.i.|)" by A1,Def8
      .= (h |^ |.i.| * (g |^ |.i.|))" by A1,Lm11
      .= (g |^ |.i.|)" * (h |^ |.i.|)" by Th16
      .= g |^ i * (h |^ |.i.|)" by A4,Def8
      .= g |^ i * (h |^ i) by A4,Def8;
  end;
end;
