
theorem Cantor5Div: :: from Cantor Theorem 5, p. 121
  for f being increasing natural-valued Arithmetic_Progression
    st for i being Nat st i < 10 holds f.i is odd Prime holds
      210 divides difference f
  proof
    let f be increasing natural-valued Arithmetic_Progression;
    assume
A1: for i being Nat st i < 10 holds f.i is odd Prime;
    reconsider m = f.0 as Nat;
    difference f = f.1 - f.0 by NUMBER06:def 5; then
    reconsider r = difference f as Nat;
A2: f = ArProg (m,r) by NUMBER06:6;
    set n = 10;
a3: for i being Nat st i < n holds
      ArProg (m,r).i is odd prime by A1,A2; then
T1: 2 divides r by XPRIMES1:2,SierpinskiTh5;
    3 divides r by a3,XPRIMES1:3,SierpinskiTh5; then
T5: 2 * 3 divides r by T1,PEPIN:4,INT_2:30,XPRIMES1:2,3;
T3: 5 divides r by a3,XPRIMES1:5,SierpinskiTh5;
    7 divides r by a3,XPRIMES1:7,SierpinskiTh5; then
T6: 5 * 7 divides r by PEPIN:4,T3,INT_2:30,XPRIMES1:5,7;
    2,5 are_coprime & 2,7 are_coprime by INT_2:30,XPRIMES1:2,5,7; then
T8: 2,5*7 are_coprime by EULER_1:14;
    3,5 are_coprime & 3,7 are_coprime by INT_2:30,XPRIMES1:3,5,7; then
    3,5*7 are_coprime by EULER_1:14; then
    35 * 6 divides r by T5,T6,PEPIN:4,T8,EULER_1:14;
    hence thesis;
  end;
