
theorem Th19:
  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;
A1: pow(x, -1) = (x |^ (|.-1 .|))" by Def2;
  assume
A2: x <> 0.L;
  then x |^ (|.-1 .|) <> 0.L by Th1;
  then
A3: pow(x, -1) <> 0.L by A1,VECTSP_1:25;
A4: 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 A2,Th1;
    end;
    suppose
A5:   -j < 0;
A6:   x |^ (|.-j.|) <> 0.L by A2,Th1;
      pow(x, -j) = (x |^ (|.-j.|))" by A5,Def2;
      hence thesis by A6,VECTSP_1:25;
    end;
  end;
A7: 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 A2,Th1;
    end;
    suppose
A8:   j+1 < 0;
A9:   x |^ (|.j+1 .|) <> 0.L by A2,Th1;
      pow(x, j+1) = (x |^ (|.j+1 .|))" by A8,Def2;
      hence thesis by A9,VECTSP_1:25;
    end;
  end;
A10: now
    per cases by Lm1;
    suppose
A11:  j >= 0;
      then reconsider n = j as Element of NAT by INT_1:3;
A12:  n + 1 = |.j + 1 .| by ABSVALUE:def 1;
      pow(x, |.j.|) = x |^ (|.j.|) by Def2;
      then
A13:  pow(x, |.j.|) <> 0.L by A2,Th1;
      pow(x, |.j+1 .|) = x |^ (|.j+1 .|) by Def2;
      then
A14:  pow(x, |.j+1 .|) <> 0.L by A2,Th1;
      n + 1 >= 0;
      hence
      pow(x, j + 1) * (pow(x, -1) * pow(x, -j)) = pow(x, |.j+1 .|) * (pow
      (x, -1) * pow(x, -j)) by ABSVALUE:def 1
        .= pow(x, |.j+1 .|) * (x" * pow(x, -j)) by Th15
        .= pow(x, |.j+1 .|) * (x" * (pow(x, j))") by A2,Th18
        .= pow(x, |.j+1 .|) * (x" * (pow(x, |.j.|))") by A11,ABSVALUE:def 1
        .= pow(x, |.j+1 .|) * ((x * pow(x, |.j.|))") by A2,A13,Th2
        .= pow(x, |.j+1 .|) * (pow(x, |.j.| + 1))" by Th17
        .= pow(x, |.j+1 .|) * (pow(x, |.j+1 .|))" by A12,ABSVALUE:def 1
        .= 1.L by A14,VECTSP_1:def 10;
    end;
    suppose
A15:  j < - 1;
A16:  pow(x, -j) <> 0.L
      proof
        reconsider k = -j as Element of NAT by A15,INT_1:3;
        pow(x, -j) = x |^ k by Def2;
        hence thesis by A2,Th1;
      end;
      pow(x, |.j+1 .|) = x |^ (|.j+1 .|) by Def2;
      then
A17:  pow(x, |.j+1 .|) <> 0.L by A2,Th1;
A18:  j + 1 < - 1 + 1 by A15,XREAL_1:6;
      hence pow(x, j+1) * (pow(x, -1) * pow(x, -j)) = (pow(x, |.j+1 .|))" * (
      pow(x, -1) * pow(x, -j)) by Lm3
        .= (pow(x, |.j+1 .|))" * (x" * pow(x, -j)) by Th15
        .= (pow(x, |.j+1 .|))" * x" * pow(x, -j) by GROUP_1:def 3
        .= (pow(x, |.j+1 .|) * x)" * pow(x, -j) by A2,A17,Th2
        .= (pow(x, (|.j + 1 .| + 1)))" * pow(x, -j) by Th17
        .= (pow(x, (- (j + 1) + 1)))" * pow(x, -j) by A18,ABSVALUE:def 1
        .= 1.L by A16,VECTSP_1:def 10;
    end;
    suppose
A19:  j = - 1;
A20:  x" <> 0.L by A2,VECTSP_1:25;
      thus pow(x, j+1) * (pow(x, -1) * pow(x, -j)) = 1.L * (pow(x, -1) * pow(x
      , -j)) by A19,Th13
        .= (pow(x, -1) * pow(x, -j))
        .= x" * pow(x, -j) by Th15
        .= x" * (pow(x, j))" by A2,Th18
        .= x" * (x")" by A19,Th15
        .= 1.L by A20,VECTSP_1:def 10;
    end;
  end;
A21: 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 A2,Th1;
    end;
    suppose
A22:  j+1 < 0;
A23:  x |^ (|.j+1 .|) <> 0.L by A2,Th1;
      pow(x, j+1) = (x |^ (|.j+1 .|))" by A22,Def2;
      hence thesis by A23,VECTSP_1:25;
    end;
  end;
  pow(x, j+1) * pow(x, -(j+1)) = pow(x, j+1) * (pow(x, j+1))" by A2,Th18
    .= 1.L by A7,VECTSP_1:def 10;
  then
A24: pow(x, -(j + 1)) = pow(x, -1) * pow(x, -j) by A10,A21,VECTSP_1:5;
  thus pow(x, j+1) = pow(x, -(-(j+1)))
    .= (pow(x, -1) * pow(x, -j))" by A2,A24,Th18
    .= (pow(x, -j))" * (pow(x, -1))" by A4,A3,Th2
    .= pow(x, -(-j)) * (pow(x, -1))" by A2,Th18
    .= pow(x, j) * pow(x, -(- 1)) by A2,Th18
    .= pow(x, j) * pow(x, 1);
end;
