reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;

theorem Th24:
  MUL_MOD(i,j,n) is_expressible_by n
proof
  set CV = (ChangeVal_1(i,n)*ChangeVal_1(j,n)) mod (2 to_power(n)+1);
  (ChangeVal_1(i,n)*ChangeVal_1(j,n)) mod (2 to_power(n)+1) < (2 to_power(
  n)+1) by NAT_D:1;
  then
A1: (ChangeVal_1(i,n)*ChangeVal_1(j,n)) mod (2 to_power(n)+1) <= 2 to_power(
  n) by NAT_1:13;
A2: 0 < 2 to_power(n) by POWER:34;
  now
    per cases by A1,XXREAL_0:1;
    suppose
      CV = 2 to_power(n);
      then ChangeVal_2(CV,n) = 0 by Def8;
      hence thesis by A2;
    end;
    suppose
A3:   CV < 2 to_power(n);
      then ChangeVal_2(CV,n) = CV by Def8;
      hence thesis by A3;
    end;
  end;
  hence thesis;
end;
