reserve n for Nat,
  i for Integer,
  p, x, x0, y for Real,
  q for Rational,
  f for PartFunc of REAL,REAL;

theorem Th1:
  for m,n be Nat holds x #Z (n+m) = (x #Z n) * (x #Z m)
proof
  let m,n be Nat;
  per cases;
  suppose
    x <> 0;
    hence thesis by PREPOWER:44;
  end;
  suppose
A1: x = 0;
    thus thesis
    proof
A2:   0 #Z (n+m) = 0|^|.n+m.| by PREPOWER:def 3
        .= 0|^(n+m) by ABSVALUE:def 1
        .= 0|^n*0|^m by NEWTON:8;
      per cases;
      suppose
A3:     n = 0 & m = 0;
        x #Z (0+0) = 1 * (x #Z 0) .= (x #Z 0) * (x #Z 0) by PREPOWER:34;
        hence thesis by A3;
      end;
      suppose
        n <> 0;
        then
A4:     0+1 <= n by NAT_1:13;
A5:     (0 #Z n) *(0 #Z m) = (0 |^ |.n.|) * (0 #Z m) by PREPOWER:def 3
          .= (0 |^ n) * (0 #Z m) by ABSVALUE:def 1
          .= 0 * (0 #Z m) by A4,NEWTON:11;
        0 #Z (n+m) = 0*0|^m by A2,A4,NEWTON:11;
        hence thesis by A1,A5;
      end;
      suppose
        m <> 0;
        then
A6:     0+1 <= m by NAT_1:13;
A7:     (0 #Z n) *(0 #Z m) = (0 |^ |.m.|) * (0 #Z n) by PREPOWER:def 3
          .= (0 |^ m) * (0 #Z n) by ABSVALUE:def 1
          .= 0 * (0 #Z n) by A6,NEWTON:11;
        0 #Z (n+m) = 0*0|^n by A2,A6,NEWTON:11;
        hence thesis by A1,A7;
      end;
    end;
  end;
end;
