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;

theorem Th53:
  n satisfies_Sierpinski_problem_86 implies n is even
  proof
    assume
A1: n satisfies_Sierpinski_problem_86;
    assume n is odd;
    then
A2: n-1 is even & n+1 is even;
    n-1 is prime or n+1 is prime by A1,Th52;
    then n-1 = 2 or n+1 = 2 by A2;
    then n = 3 or n = 1;
    hence thesis by A1,Th50;
  end;
