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 Th50:
  n satisfies_Sierpinski_problem_86 implies n >= 6
  proof
    given a,b,c being Prime such that
A1: a,b,c are_mutually_distinct and
A2: n|^2-1 = a*b*c;
    a >= 2 & b >= 3 & c >= 5 or a >= 2 & b >= 5 & c >= 3 or
    a >= 3 & b >= 2 & c >= 5 or a >= 3 & b >= 5 & c >= 2 or
    a >= 5 & b >= 2 & c >= 3 or a >= 5 & b >= 3 & c >= 2 by A1,Th49;
    then
A3: a*b*c >= 2*3*5 or a*b*c >= 3*5*2 or a*b*c >= 5*2*3 by Lm15;
A4: 5 = 6-1;
A5: n|^2 = n*n by WSIERP_1:1;
    assume n < 6;
    then n = 0 or ... or n = 5 by A4,NUMBER02:7;
    hence thesis by A2,A3,A5;
  end;
