reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem
  1 < a <= 100 implies
  ex n being positive Nat st n <= 6 & a|^(2|^n) + 1 is composite
  proof
    assume that
A1: 1 < a and
A2: a <= 100;
    deffunc G(Nat) = a|^(2|^$1);
    deffunc F(Nat) = G($1) + 1;
    per cases by A1,A2,Lm28;
    suppose
A3:   a is odd;
      take n = 1;
      thus n <= 6;
      0+1 <= G(n) by A1,NAT_1:13;
      then 1+1 <= F(n) by XREAL_1:6;
      hence 2 <= F(n);
A4:   F(n) is even by A3;
      a|^2 = a*a by WSIERP_1:1;
      then F(n) <> 2 by A1,NAT_1:15;
      hence F(n) is non prime by A4;
    end;
    suppose
A5:   a = 2;
      take n = 5;
      thus n <= 6;
      thus 2 <= F(n) by A5,Lm51,Lm50;
      thus F(n) is non prime by A5,NAT_6:44;
    end;
    suppose
A6:   a = 4;
      take n = 4;
      thus n <= 6;
      thus 2 <= F(n) by A6,Lm46,Lm51,Lm50;
      thus F(n) is non prime by A6,Lm46,NAT_6:44;
    end;
    suppose
A7:   a = 6;
      take n = 3;
      thus n <= 6;
A8:   17 < F(n) by A7,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+6)|^(2|^3)+1 by Th37;
      hence thesis by A7,A8;
    end;
    suppose
A9:   a = 8;
      take n = 2;
      thus n <= 6;
A10:  17 < F(n) by A9,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+8)|^(2|^2)+1 by Th38;
      hence thesis by A9,A10;
    end;
    suppose
A11:  a = 10;
      take n = 3;
      thus n <= 6;
A12:  17 < F(n) by A11,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+10)|^(2|^3)+1 by Th39;
      hence thesis by A11,A12;
    end;
    suppose
A13:  a = 12;
      take n = 3;
      thus n <= 6;
A14:  17 < F(n) by A13,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+12)|^(2|^3)+1 by Th40;
      hence thesis by A13,A14;
    end;
    suppose
A15:  a = 14;
      take n = 3;
      thus n <= 6;
A16:  17 < F(n) by A15,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+14)|^(2|^3)+1 by Th41;
      hence thesis by A15,A16;
    end;
    suppose
A17:  a = 16;
      take n = 3;
      thus n <= 6;
      thus 2 <= F(n) by A17,Lm51,Lm50,Lm47;
      thus F(n) is non prime by A17,Lm47,NAT_6:44;
    end;
    suppose
A18:  a = 18;
      take n = 1;
      thus n <= 6;
A19:  G(n) = 18*18 by A18,WSIERP_1:1;
      hence 2 <= F(n);
      325 = 5*65;
      then 5 divides 325;
      hence F(n) is non prime by A19;
    end;
    suppose
A20:  a = 20;
      take n = 3;
      thus n <= 6;
A21:  17 < F(n) by A20,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+20)|^(2|^3)+1 by Th42;
      hence thesis by A20,A21;
    end;
    suppose
A22:  a = 22;
      take n = 3;
      thus n <= 6;
A23:  17 < F(n) by A22,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+22)|^(2|^3)+1 by Th43;
      hence thesis by A22,A23;
    end;
    suppose
A24:  a = 24;
      take n = 3;
      thus n <= 6;
A25:  17 < F(n) by A24,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+24)|^(2|^3)+1 by Th44;
      hence thesis by A24,A25;
    end;
    suppose
A26:  a = 26;
      take n = 2;
      thus n <= 6;
A27:  17 < F(n) by A26,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+26)|^(2|^2)+1 by Th45;
      hence thesis by A26,A27;
    end;
    suppose
A28:  a = 28;
      take n = 3;
      thus n <= 6;
A29:  17 < F(n) by A28,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+28)|^(2|^3)+1 by Th46;
      hence thesis by A28,A29;
    end;
    suppose
A30:  a = 30;
      take n = 1;
      thus n <= 6;
A31:  17 < F(n) by A30,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+30)|^(2|^1)+1 by Th47;
      hence thesis by A30,A31;
    end;
    suppose
A32:  a = 32;
      take n = 2;
      thus n <= 6;
A33:  17 < F(n) by A32,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*0+32)|^(2|^2)+1 by Th48;
      hence thesis by A32,A33;
    end;
    suppose
A34:  a = 34;
      take n = 1;
      thus n <= 6;
A35:  G(n) = 34*34 by A34,WSIERP_1:1;
      hence 2 <= F(n);
      1157 = 13*89;
      then 13 divides 1157;
      hence F(n) is non prime by A35;
    end;
    suppose
A36:  a = 36;
      take n = 2;
      thus n <= 6;
A37:  17 < F(n) by A36,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+2)|^(2|^2)+1 by Th35;
      hence thesis by A36,A37;
    end;
    suppose
A38:  a = 38;
      take n = 1;
      thus n <= 6;
A39:  17 < F(n) by A38,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+4)|^(2|^1)+1 by Th36;
      hence thesis by A38,A39;
    end;
    suppose
A40:  a = 40;
      take n = 3;
      thus n <= 6;
A41:  17 < F(n) by A40,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+6)|^(2|^3)+1 by Th37;
      hence thesis by A40,A41;
    end;
    suppose
A42:  a = 42;
      take n = 2;
      thus n <= 6;
A43:  17 < F(n) by A42,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+8)|^(2|^2)+1 by Th38;
      hence thesis by A42,A43;
    end;
    suppose
A44:  a = 44;
      take n = 3;
      thus n <= 6;
A45:  17 < F(n) by A44,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+10)|^(2|^3)+1 by Th39;
      hence thesis by A44,A45;
    end;
    suppose
A46:  a = 46;
      take n = 3;
      thus n <= 6;
A47:  17 < F(n) by A46,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+12)|^(2|^3)+1 by Th40;
      hence thesis by A46,A47;
    end;
    suppose
A48:  a = 48;
      take n = 3;
      thus n <= 6;
A49:  17 < F(n) by A48,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+14)|^(2|^3)+1 by Th41;
      hence thesis by A48,A49;
    end;
    suppose
A50:  a = 50;
      take n = 1;
      thus n <= 6;
A51:  G(n) = 50*50 by A50,WSIERP_1:1;
      hence 2 <= F(n);
      2501 = 41*61;
      then 41 divides 2501;
      hence F(n) is non prime by A51;
    end;
    suppose
A52:  a = 52;
      take n = 1;
      thus n <= 6;
A53:  G(n) = 52*52 by A52,WSIERP_1:1;
      hence 2 <= F(n);
      2705 = 5*541;
      then 5 divides 2705;
      hence F(n) is non prime by A53;
    end;
    suppose
A54:  a = 54;
      take n = 3;
      thus n <= 6;
A55:  17 < F(n) by A54,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+20)|^(2|^3)+1 by Th42;
      hence thesis by A54,A55;
    end;
    suppose
A56:  a = 56;
      take n = 3;
      thus n <= 6;
A57:  17 < F(n) by A56,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+22)|^(2|^3)+1 by Th43;
      hence thesis by A56,A57;
    end;
    suppose
A58:  a = 58;
      take n = 3;
      thus n <= 6;
A59:  17 < F(n) by A58,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+24)|^(2|^3)+1 by Th44;
      hence thesis by A58,A59;
    end;
    suppose
A60:  a = 60;
      take n = 2;
      thus n <= 6;
A61:  17 < F(n) by A60,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+26)|^(2|^2)+1 by Th45;
      hence thesis by A60,A61;
    end;
    suppose
A62:  a = 62;
      take n = 3;
      thus n <= 6;
A63:  17 < F(n) by A62,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+28)|^(2|^3)+1 by Th46;
      hence thesis by A62,A63;
    end;
    suppose
A64:  a = 64;
      take n = 1;
      thus n <= 6;
A65:  17 < F(n) by A64,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+30)|^(2|^1)+1 by Th47;
      hence thesis by A64,A65;
    end;
    suppose
A66:  a = 66;
      take n = 2;
      thus n <= 6;
A67:  17 < F(n) by A66,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*1+32)|^(2|^2)+1 by Th48;
      hence thesis by A66,A67;
    end;
    suppose
A68:  a = 68;
      take n = 1;
      thus n <= 6;
A69:  G(n) = 68*68 by A68,WSIERP_1:1;
      hence 2 <= F(n);
      4625 = 5*925;
      then 5 divides 4625;
      hence F(n) is non prime by A69;
    end;
    suppose
A70:  a = 70;
      take n = 2;
      thus n <= 6;
A71:  17 < F(n) by A70,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+2)|^(2|^2)+1 by Th35;
      hence thesis by A70,A71;
    end;
    suppose
A72:  a = 72;
      take n = 1;
      thus n <= 6;
A73:  17 < F(n) by A72,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+4)|^(2|^1)+1 by Th36;
      hence thesis by A72,A73;
    end;
    suppose
A74:  a = 74;
      take n = 3;
      thus n <= 6;
A75:  17 < F(n) by A74,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+6)|^(2|^3)+1 by Th37;
      hence thesis by A74,A75;
    end;
    suppose
A76:  a = 76;
      take n = 2;
      thus n <= 6;
A77:  17 < F(n) by A76,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+8)|^(2|^2)+1 by Th38;
      hence thesis by A76,A77;
    end;
    suppose
A78:  a = 78;
      take n = 3;
      thus n <= 6;
A79:  17 < F(n) by A78,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+10)|^(2|^3)+1 by Th39;
      hence thesis by A78,A79;
    end;
    suppose
A80:  a = 80;
      take n = 3;
      thus n <= 6;
A81:  17 < F(n) by A80,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+12)|^(2|^3)+1 by Th40;
      hence thesis by A80,A81;
    end;
    suppose
A82:  a = 82;
      take n = 3;
      thus n <= 6;
A83:  17 < F(n) by A82,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+14)|^(2|^3)+1 by Th41;
      hence thesis by A82,A83;
    end;
    suppose
A84:  a = 84;
      take n = 6;
      thus n <= 6;
A85:  84|^(2|^6) = 84|^2|^32 by Lm6,NEWTON:9;
A86:  84|^2 = 84*84 by WSIERP_1:1;
      then 84|^2|^1 < 84|^2|^32 by PEPIN:66;
      then 256 < 84|^(2|^6) by A85,A86,XXREAL_0:2;
      then
A87:  256+1 < F(n) by A84,XREAL_1:8;
      hence 2 <= F(n) by XXREAL_0:2;
      thus thesis by A87,A84,Lm45;
    end;
    suppose
A88:  a = 86;
      take n = 1;
      thus n <= 6;
A89:  G(n) = 86*86 by A88,WSIERP_1:1;
      hence 2 <= F(n);
      7397 = 13*569;
      then 13 divides 7397;
      hence F(n) is non prime by A89;
    end;
    suppose
A90:  a = 88;
      take n = 3;
      thus n <= 6;
A91:  17 < F(n) by A90,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+20)|^(2|^3)+1 by Th42;
      hence thesis by A90,A91;
    end;
    suppose
A92:  a = 90;
      take n = 3;
      thus n <= 6;
A93:  17 < F(n) by A92,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+22)|^(2|^3)+1 by Th43;
      hence thesis by A92,A93;
    end;
    suppose
A94:  a = 92;
      take n = 3;
      thus n <= 6;
A95:  17 < F(n) by A94,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+24)|^(2|^3)+1 by Th44;
      hence thesis by A94,A95;
    end;
    suppose
A96:  a = 94;
      take n = 2;
      thus n <= 6;
A97:  17 < F(n) by A96,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+26)|^(2|^2)+1 by Th45;
      hence thesis by A96,A97;
    end;
    suppose
A98:  a = 96;
      take n = 3;
      thus n <= 6;
A99:  17 < F(n) by A98,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+28)|^(2|^3)+1 by Th46;
      hence thesis by A98,A99;
    end;
    suppose
A100: a = 98;
      take n = 1;
      thus n <= 6;
A101: 17 < F(n) by A100,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+30)|^(2|^1)+1 by Th47;
      hence thesis by A100,A101;
    end;
    suppose
A102: a = 100;
      take n = 2;
      thus n <= 6;
A103: 17 < F(n) by A102,Lm49;
      hence 2 <= F(n) by XXREAL_0:2;
      17 divides (34*2+32)|^(2|^2)+1 by Th48;
      hence thesis by A102,A103;
    end;
  end;
