
theorem :: Problem 72
  for f being increasing natural-valued Arithmetic_Progression
    st for i being Nat st i < 10 holds f.i is odd Prime holds
      f.9 >= 2089
  proof
    let f be increasing natural-valued Arithmetic_Progression;
    assume
A1: for i being Nat st i < 10 holds f.i is odd Prime; then
A3: difference f >= 210 by NAT_D:7,Cantor5Div;
    per cases;
    suppose
      difference f <= 210; then
      difference f = 210 by A3,XXREAL_0:1; then
A2:   f.0 >= 199 by A1,Theorem72V1;
a4:   f = ArProg (f.0, difference f) by NUMBER06:6;
      9 * difference f >= 9 * 210 by A1,Cantor5,XREAL_1:64; then
      f.0 + 9 * difference f >= 199 + 9 * 210 by A2,XREAL_1:7;
      hence thesis by a4,NUMBER06:7;
    end;
    suppose
T1:   difference f > 210;
T2:   difference f = 210 * ((difference f) div 210)
        by A1,Cantor5Div,NAT_D:3;
      (difference f) div 210 > 1
      proof
        assume (difference f) div 210 <= 1; then
        (difference f) div 210 < 1 + 1 by NAT_1:13; then
        (difference f) div 210 = 1 or (difference f) div 210 = 0
           by NAT_1:23;
        hence thesis by T1,T2;
      end; then
      (difference f) div 210 >= 1 + 1 by NAT_1:13; then
S0:   difference f >= 2 * 210 by T2,XREAL_1:64;
      f.0 is Prime by A1; then
      f.0 > 1 by INT_2:def 4; then
S1:   f.0 >= 1+1 by NAT_1:13;
      f = ArProg (f.0, difference f) by NUMBER06:6; then
A4:   f.9 = 9 * difference f + f.0 by NUMBER06:7;
      9 * difference f >= 9 * (2 * 210) by S0,XREAL_1:64; then
      f.0 + 9 * difference f >= 2 + 9 * 420 by S1,XREAL_1:7;
      hence thesis by A4,XXREAL_0:2;
    end;
  end;
