reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th103:
  n is odd & i,j are_coprime & i*j = z|^n implies ex k be Integer st i = k|^n
  proof
    assume
A1: n is odd;
    assume i,j are_coprime;
    then
A2: |.i.|,|.j.| are_coprime by INT_2:34;
    assume
A3: i*j = z|^n;
A4: |.i.|*|.j.| = |.i*j.| by COMPLEX1:65;
    |.z|^n.| = |.z.||^n by TAYLOR_2:1;
    then consider k being Nat such that
A5: k|^n = |.i.| by A2,A3,A4,NEWTON02:30;
    per cases;
    suppose
A6:   i >= 0;
      take k;
      thus k|^n = i by A5,A6,ABSVALUE:def 1;
    end;
    suppose i <= 0;
      then
A7:   |.i.| = -i by ABSVALUE:30;
      take -k;
      thus (-k)|^n = -(k|^n) by A1,POWER:2
      .= i by A5,A7;
    end;
  end;
