reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th43:
  for m,n being Nat holds a<>0 implies a #Z (m-n) = a |^ m / a |^ n
proof
  let m,n be Nat;
  assume
A1: a<>0;
  per cases;
  suppose
    m-n>=0;
    then reconsider m1 = m-n as Element of NAT by INT_1:3;
A2: a #Z (m-n) * a |^ n = a |^ m1 * a |^ n by Th36
      .= a |^ (m1+n) by NEWTON:8
      .= a |^ m;
    a |^ n <> 0 by A1,Th5;
    hence thesis by A2,XCMPLX_1:89;
  end;
  suppose
    m-n<0;
    then -(m-n)>0;
    then reconsider m1 = n-m as Element of NAT by INT_1:3;
A3: a #Z (n-m) = a #Z (-(m-n)) .= 1/a #Z (m-n) by Th41;
    a #Z (n-m) * a |^ m = a |^ m1 * a |^ m by Th36
      .= a |^ (m1+m) by NEWTON:8
      .= a |^ n;
    then
A4: a |^ m / a #Z (m-n) = a |^ n by A3;
    a |^ n <> 0 by A1,Th5;
    hence thesis by A4,XCMPLX_1:54;
  end;
end;
