
theorem Diff210Mod11:
  for f being increasing natural-valued Arithmetic_Progression
    st (for i being Nat st i < 10 holds f.i is odd Prime) &
    difference f = 210 holds
       for f0 being Nat st f0 = f.0 holds f0 mod 11 = 1
  proof
    let f be increasing natural-valued Arithmetic_Progression;
    assume
A1: for i being Nat st i < 10 holds f.i is odd Prime;
    assume
B1: 210 = difference f;
    let f0 be Nat;
AA: f = ArProg (f.0,difference f) by NUMBER06:6;
    assume
A2: f0 = f.0; then
A3: f0 is Prime by A1; then
    f0 > 1 by INT_2:def 4; then
R1: f0 >= 1+1 by NAT_1:13;
J1: f0 mod 11 <> 0
    proof
      assume f0 mod 11 = 0; then
      11 divides f0 by PEPIN:6; then
      f0 = 1 or f0 = 11 by A3,INT_2:def 4; then
a3:   f.1 = 11 + 1 * 210 by A3,INT_2:def 4,A2,AA,NUMBER06:7,B1
         .= 221;
      13 divides 13 * 17; then
      221 is not Prime by INT_2:def 4;
      hence thesis by A1,a3;
    end;
JA: f0 mod 11 <> 10
    proof
      assume
K1:   f0 mod 11 = 10;
      f0 + 210 >= 210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 210 <> 11;
K2:   210 mod 11 = (100 + 11 * 10) mod 11
                .= (1 + 9 * 11) mod 11 by NAT_D:21
                .= 1 mod 11 by NAT_D:21
                .= 1 by NAT_D:24;
      (f0 + 210) mod 11 = ((f0 mod 11) + (210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by K2,K1,NAT_D:25; then
      11 divides (f0 + 210) by PEPIN:6; then
K3:   f0 + 210 is not Prime by K0,INT_2:def 4;
      difference f = f.1 - f.0 by NUMBER06:def 5;
      hence thesis by K3,A1,A2,B1;
    end;
JC: f0 mod 11 <> 9
    proof
      assume
K1:   f0 mod 11 = 9;
      f0 + 2*210 >= 2*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 2*210 <> 11;
k2:   2*210 mod 11 = (210 + 100 + 11 * 10) mod 11
                  .= (210 + 1 + 9 * 11) mod 11 by NAT_D:21
                  .= (19 * 11 + 2) mod 11 by NAT_D:21
                  .= 2 mod 11 by NAT_D:21
                  .= 2 by NAT_D:24;
      (f0 + 2*210) mod 11 = ((f0 mod 11) + (2*210 mod 11)) mod 11 by NAT_D:66
                         .= 0 by K1,k2,NAT_D:25; then
      11 divides (f0 + 2*210) by PEPIN:6; then
K3:   f0 + 2*210 is not Prime by K0,INT_2:def 4;
      f.2 = f.0 + 2 * difference f by AA,NUMBER06:7;
      hence thesis by A1,K3,A2,B1;
    end;
JD: f0 mod 11 <> 8
    proof
      assume
K1:   f0 mod 11 = 8;
      f0 + 3*210 >= 3*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 3*210 <> 11;
k2:   3*210 mod 11 = (210+210 + 100 + 11 * 10) mod 11
                .= (210 + 210 + 1 + 9 * 11) mod 11 by NAT_D:21
                .= (212 + 19*11) mod 11 by NAT_D:21
                .= (100 + 2 + 11*10) mod 11 by NAT_D:21
                .= (1 + 2 + 9*11) mod 11 by NAT_D:21
                .= (1 + 2) mod 11 by NAT_D:21
                .= 3 by NAT_D:24;
      (f0 + 3*210) mod 11 = ((f0 mod 11) + (3*210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by K1,k2,NAT_D:25; then
      11 divides (f0 + 3*210) by PEPIN:6; then
K3:   f0 + 3*210 is not Prime by K0,INT_2:def 4;
      f.3 = f.0 + 3 * difference f by AA,NUMBER06:7;
      hence thesis by K3,A1,A2,B1;
    end;
JB: f0 mod 11 <> 7
    proof
      assume
K1:   f0 mod 11 = 7;
      f0 + 4*210 >= 4*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 4*210 <> 11;
k2:   4*210 mod 11 = (4+76*11) mod 11
                .= 4 mod 11 by NAT_D:21
                .= 4 by NAT_D:24;
      (f0 + 4*210) mod 11 = ((f0 mod 11) + (4*210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by K1,k2,NAT_D:25; then
      11 divides (f0 + 4*210) by PEPIN:6; then
K3:   f0 + 4*210 is not Prime by K0,INT_2:def 4;
      f.4 = f.0 + 4 * difference f by AA,NUMBER06:7;
      hence thesis by K3,A1,A2,B1;
    end;
JE: f0 mod 11 <> 6
    proof
      assume
K1:   f0 mod 11 = 6;
      f0 + 5*210 >= 5*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 5*210 <> 11;
k2:   5*210 mod 11 = (5+95*11) mod 11
                .= 5 mod 11 by NAT_D:21
                .= 5 by NAT_D:24;
      (f0 + 5*210) mod 11 = ((f0 mod 11) + (5*210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by K1,k2,NAT_D:25; then
      11 divides (f0 + 5*210) by PEPIN:6; then
K3:   f0 + 5*210 is not Prime by K0,INT_2:def 4;
      f.5 = f.0 + 5 * 210 by AA,B1,NUMBER06:7;
      hence thesis by K3,A1,A2;
    end;
JF: f0 mod 11 <> 5
    proof
      assume
K1:   f0 mod 11 = 5;
      f0 + 6*210 >= 6*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 6*210 <> 11;
k2:   6 * 210 mod 11 = (6 + 114 * 11) mod 11
                .= 6 mod 11 by NAT_D:21
                .= 6 by NAT_D:24;
      (f0 + 6*210) mod 11 = ((f0 mod 11) + (6*210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by K1,k2,NAT_D:25; then
      11 divides (f0 + 6*210) by PEPIN:6; then
K3:   f0 + 6 * 210 is not Prime by K0,INT_2:def 4;
      f.6 = f.0 + 6 * difference f by AA,NUMBER06:7;
      hence thesis by K3,A1,A2,B1;
    end;
JG: f0 mod 11 <> 4
    proof
      assume
K1:   f0 mod 11 = 4;
      f0 + 7*210 >= 7*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 7*210 <> 11;
k2:   7*210 mod 11 = (7 + 133 * 11) mod 11
                .= 7 mod 11 by NAT_D:21
                .= 7 by NAT_D:24;
      (f0 + 7*210) mod 11 = ((f0 mod 11) + (7*210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by K1,k2,NAT_D:25; then
      11 divides (f0 + 7*210) by PEPIN:6; then
K3:   f0 + 7*210 is not Prime by K0,INT_2:def 4;
      f.7 = f.0 + 7 * difference f by AA,NUMBER06:7;
      hence thesis by K3,A1,A2,B1;
    end;
JH: f0 mod 11 <> 3
    proof
      assume
K1:   f0 mod 11 = 3;
      f0 + 8*210 >= 8*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 8*210 <> 11;
k2:   8*210 mod 11 = (8 + 152 * 11) mod 11
                .= 8 mod 11 by NAT_D:21
                .= 8 by NAT_D:24;
      (f0 + 8*210) mod 11 = ((f0 mod 11) + (8*210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by K1,k2,NAT_D:25; then
      11 divides (f0 + 8*210) by PEPIN:6; then
K3:   f0 + 8*210 is not Prime by K0,INT_2:def 4;
      f.8 = f.0 + 8 * difference f by AA,NUMBER06:7;
      hence thesis by K3,A1,A2,B1;
    end;
JI: f0 mod 11 <> 2
    proof
      assume
K1:   f0 mod 11 = 2;
      f0 + 9*210 >= 9*210 + 2 by R1,XREAL_1:6; then
K0:   f0 + 9*210 <> 11;
k2:   9*210 mod 11 = (9+171*11) mod 11
                .= 9 mod 11 by NAT_D:21
                .= 9 by NAT_D:24;
      (f0 + 9*210) mod 11 = ((f0 mod 11) + (9*210 mod 11)) mod 11 by NAT_D:66
                       .= 0 by NAT_D:25,K1,k2; then
      11 divides (f0 + 9*210) by PEPIN:6; then
K3:   f0 + 9*210 is not Prime by K0,INT_2:def 4;
      f.9 = f.0 + 9 * difference f by AA,NUMBER06:7;
      hence thesis by K3,A1,A2,B1;
    end;
    f0 mod (10 + 1) = 0 or ... or f0 mod (10 + 1) = 10
       by NUMBER03:11;
::: repeated in NUMBER05
    hence thesis by JD,JE,JF,JG,JH,JI,J1,JA,JB,JC;
  end;
