reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  n > 1 implies 1/5 * (2|^(4*n+2)+1) is composite
  proof
    assume
A1: n > 1;
    set W = 2|^(4*n+2)+1;
    set L = 2|^(2*n+1)-2|^(n+1)+1;
    set R = 2|^(2*n+1)+2|^(n+1)+1;
A2: 2|^(2*n+1)*2|^(2*n+1) = 2|^((2*n+1)+(2*n+1)) by NEWTON:8;
A3: 2*2|^(2*n+1) = 2|^(2*n+1+1) by NEWTON:6;
A4: 2|^(n+1)*2|^(n+1) = 2|^(n+1+(n+1)) by NEWTON:8;
    2|^(2*n+1+1) = 2|^(n+1+(n+1));
    then
A5: 2|^(4*n+2) = 2|^(2*n+1)*2|^(2*n+1) + 2*2|^(2*n+1) - 2|^(n+1)*2|^(n+1)
    by A2,A3,A4;
A6: 2|^(n+1+n) = 2|^(n+1)*2|^n by NEWTON:8;
A7: n >= 1+1 by A1,NAT_1:13;
    then n+1 >= 2+1 by XREAL_1:6;
    then
A8: 2|^(n+1) >= 2|^3 by PREPOWER:93;
A9: 2|^n >= 2|^2 by A7,PREPOWER:93;
    then 2|^n-1 >= 4-1 by Lm1,XREAL_1:9;
    then
A10: 2|^(n+1)*(2|^n-1) >= 8*3 by A8,Lm2,XREAL_1:66;
    then
A11: 2|^(n+1)*(2|^n-1)+1 >= 24+1 by XREAL_1:6;
    then L*R >= 25*R by A6,XREAL_1:64;
    then
A12: 1/5*W >= 1/5 * (25*R) by A5,XREAL_1:64;
    reconsider L as Element of NAT by A6,A11;
    2|^n+1 >= 4+1 by A9,Lm1,XREAL_1:6;
    then
A13: 2|^(n+1)*(2|^n+1) >= 8*5 by A8,Lm2,XREAL_1:66;
    then
A14: 2|^(n+1)*(2|^n+1)+1 >= 40+1 by XREAL_1:6;
    then 5*R >= 5*41 by A6,XREAL_1:64;
    then 2 <= 5*R by XXREAL_0:2;
    hence 2 <= 1/5*W by A12,XXREAL_0:2;
A15: W = L*R by A5;
    then per cases by Th40,XPRIMES1:5,INT_5:7;
    suppose 5 divides L;
      then consider N being Nat such that
A16:  L = 5*N;
A17:  1/5*W = 1/5*(5*N*R) by A15,A16
      .= N*R;
A18:  now
        assume N < 2;
        then N = 0 or N = 1 by NAT_1:23;
        hence thesis by A6,A10,A16;
      end;
      R >= 2 by A6,A14,XXREAL_0:2;
      hence thesis by A17,A18,Th3;
    end;
    suppose 5 divides R;
      then consider N being Nat such that
A19:  R = 5*N;
A20:  1/5*W = 1/5*(5*N*L) by A15,A19
      .= N*L;
A21:  now
        assume N < 2;
        then N = 0 or N = 1 by NAT_1:23;
        hence thesis by A6,A13,A19;
      end;
      L >= 2 by A6,A11,XXREAL_0:2;
      hence thesis by A20,A21,Th3;
    end;
  end;
