
theorem Th26:
  for L being Field, x being Element of L st x <> 0.L for i,j
  being Integer holds pow(x,i * j) = pow(pow(x,i), j)
proof
  let L be Field, x being Element of L;
  assume
A1: x <> 0.L;
  let i,j be Integer;
  per cases;
  suppose
    i >= 0 & j >= 0;
    then reconsider m = i, n = j as Element of NAT by INT_1:3;
    thus pow(x, i*j) = pow(pow(x, m), n) by Th24
      .= pow(pow(x, i), j);
  end;
  suppose
A2: i >= 0 & j < 0;
    then reconsider m = i, n = - j as Element of NAT by INT_1:3;
A3: pow(x, i) = x |^ m by Def2;
    then
A4: pow(x, i) <> 0.L by A1,Th1;
A5: pow(pow(x, i), j) = ((pow(x, i)) |^ |.j.|)" by A2,Def2;
    i * j = - (i * n);
    hence pow(x, i*j) = (pow(x, i * n))" by A1,Th18
      .= pow(x", i*n) by A1,Th25
      .= pow(pow(x", m), n) by Th24
      .= pow((pow(x, i))", n) by A1,Th25
      .= (pow((pow(x, i))", j))" by A4,Th18,VECTSP_1:25
      .= ((pow(pow(x, i), j))")" by A1,A3,Th1,Th25
      .= pow(pow(x, i), j) by A4,A5,Th1,VECTSP_1:24;
  end;
  suppose
A6: i < 0 & j >= 0;
    then reconsider m = - i, n = j as Element of NAT by INT_1:3;
A7: pow(x, i) = (x |^ (|.i.|))" by A6,Def2;
    i * j = - (m * j);
    hence pow(x, i*j) = (pow(x, m*j))" by A1,Th18
      .= pow(x", m*j) by A1,Th25
      .= pow(pow(x", m), n) by Th24
      .= pow((pow(x", i))", n) by A1,Th18,VECTSP_1:25
      .= pow((pow(x, i))"", j) by A1,Th25
      .= pow(pow(x, i), j) by A1,A7,Th1,VECTSP_1:24;
  end;
  suppose
A8: j < 0 & i < 0;
    then reconsider m = - i, n = - j as Element of NAT by INT_1:3;
A9: pow(x, -i) = x |^ m by Def2;
    x" <> 0.L by A1,VECTSP_1:25;
    then
A10: (x") |^ (|.i.|) <> 0.L by Th1;
A11: pow(x", i) = ((x") |^ (|.i.|))" by A8,Def2;
    i * j * ((- 1) * (- 1)) = m * n;
    hence pow(x, i*j) = pow(pow(x, m), n) by Th24
      .= (pow(pow(x, -i), j))" by A1,A9,Th1,Th18
      .= (pow((pow(x, i))", j))" by A1,Th18
      .= (pow(pow(x", i), j))" by A1,Th25
      .= pow((pow(x", i))", j) by A11,A10,Th25,VECTSP_1:25
      .= pow(pow(x"", i), j) by A1,Th25,VECTSP_1:25
      .= pow(pow(x, i), j) by A1,VECTSP_1:24;
  end;
end;
