
theorem Th20:
  for L being Field, j being Integer, x being Element of L st x <>
  0.L holds pow(x,j-1) = pow(x,j) * pow(x,-1)
proof
  let L be Field;
  let j be Integer;
  let x be Element of L;
  assume
A1: x <> 0.L;
A2: pow(x, j-1) <> 0.L
  proof
    per cases;
    suppose
      0 <= j-1;
      then reconsider k = j-1 as Element of NAT by INT_1:3;
      pow(x, j-1) = x |^ k by Def2;
      hence thesis by A1,Th1;
    end;
    suppose
A3:   j-1 < 0;
A4:   x |^ (|.j-1 .|) <> 0.L by A1,Th1;
      pow(x, j-1) = (x |^ (|.j-1 .|))" by A3,Def2;
      hence thesis by A4,VECTSP_1:25;
    end;
  end;
A5: now
    per cases by Lm2;
    suppose
A6:   j >= 1;
      then
A7:   |.j.| = j by ABSVALUE:def 1;
      pow(x, |.-j.|) = x |^ (|.-j.|) by Def2;
      then
A8:   pow(x, |.-j.|) <> 0.L by A1,Th1;
A9:   |.j.| = |.- j.| by COMPLEX1:52;
      j >= 1 + 0 by A6;
      then
A10:  j - 1 >= 0 by XREAL_1:19;
      then
A11:  |.j - 1 .| + 1 = j - 1 + 1 by ABSVALUE:def 1
        .= j;
      thus pow(x, j-1) * (x * pow(x, -j)) = pow(x, j-1) * x * pow(x, -j) by
GROUP_1:def 3
        .= pow(x, |.j-1 .|) * x * pow(x, -j) by A10,ABSVALUE:def 1
        .= pow(x, |.j-1 .|) * x * ((pow(x, |.-j.|))") by A6,Lm4
        .= pow(x, |.j-1 .| + 1) * ((pow(x, |.-j.|))") by Th17
        .= 1.L by A8,A11,A7,A9,VECTSP_1:def 10;
    end;
    suppose
A12:  j < 0;
      pow(x, |.j-1 .|) = x |^ (|.j-1 .|) by Def2;
      then
A13:  pow(x, |.j-1 .|) <> 0.L by A1,Th1;
A14:  1 - j = - (j - 1);
      thus pow(x, j-1) * (x * pow(x, -j)) = (pow(x, |.j-1 .|))" * (x * pow(x,
      -j)) by A12,Lm3
        .= (pow(x, |.j-1 .|))" * (x * pow(x, |.-j.|)) by A12,ABSVALUE:def 1
        .= (pow(x, |.j-1 .|))" * pow(x, 1 + |.-j.|) by Th17
        .= (pow(x, |.j-1 .|))" * pow(x, 1 + (-j)) by A12,ABSVALUE:def 1
        .= (pow(x, |.j-1 .|))" * pow(x, |.j-1 .|) by A12,A14,ABSVALUE:def 1
        .= 1.L by A13,VECTSP_1:def 10;
    end;
    suppose
A15:  j = 0;
      hence pow(x, j-1) * (x * pow(x, -j)) = x" * (x * pow(x, -j)) by Th15
        .= x" * x * pow(x, -j) by GROUP_1:def 3
        .= 1.L * pow(x, -j) by A1,VECTSP_1:def 10
        .= pow(x, 0) by A15
        .= 1.L by Th13;
    end;
  end;
A16: pow(x, -j) <> 0.L
  proof
    per cases;
    suppose
      0 <= -j;
      then reconsider k = -j as Element of NAT by INT_1:3;
      pow(x, -j) = x |^ k by Def2;
      hence thesis by A1,Th1;
    end;
    suppose
A17:  -j < 0;
A18:  x |^ (|.-j.|) <> 0.L by A1,Th1;
      pow(x, -j) = (x |^ (|.-j.|))" by A17,Def2;
      hence thesis by A18,VECTSP_1:25;
    end;
  end;
A19: pow(x, j-1) <> 0.L
  proof
    per cases;
    suppose
      0 <= j-1;
      then reconsider k = j-1 as Element of NAT by INT_1:3;
      pow(x, j-1) = x |^ k by Def2;
      hence thesis by A1,Th1;
    end;
    suppose
A20:  j-1 < 0;
A21:  x |^ (|.j-1 .|) <> 0.L by A1,Th1;
      pow(x, j-1) = (x |^ (|.j-1 .|))" by A20,Def2;
      hence thesis by A21,VECTSP_1:25;
    end;
  end;
  pow(x, j-1) * (pow(x, 1-j)) = pow(x, j-1) * pow(x, -(j-1))
    .= pow(x, j-1) * (pow(x, j-1))" by A1,Th18
    .= 1.L by A2,VECTSP_1:def 10;
  then x * pow(x, -j) = pow(x, 1-j) by A5,A19,VECTSP_1:5;
  then (pow(x, 1-j))" = (pow(x, -j))" * x" by A1,A16,Th2
    .= pow(x, -(- j)) * x" by A1,Th18
    .= pow(x, j) * pow(x, -1) by Th15;
  then pow(x, j) * pow(x, -1) = pow(x, -(1-j)) by A1,Th18;
  hence thesis;
end;
