reserve a,b,c,k,k9,m,n,n9,p,p9 for Nat;
reserve i,i9 for Integer;
reserve X for Pythagorean_triple;

theorem
  { X: X is non degenerate & X is simplified } is infinite
proof
  set T = { X: X is non degenerate & X is simplified };
  for m ex n st n >= m & n in union T
  proof
    let m;
    set m9 = m + 1;
    set n = 4*m9;
    take n;
    consider X such that
A1: X is non degenerate & X is simplified and
A2: 4*m9 in X by Th14;
    n + 0 = 1*m9 + 3*m9;
    then
A3: n >= m9 + 0 by XREAL_1:6;
    m9 >= m by NAT_1:11;
    hence n >= m by A3,XXREAL_0:2;
    X in T by A1;
    hence thesis by A2,TARSKI:def 4;
  end;
  then
A4: union T is infinite by Th9;
  now
    let X be set;
    assume X in T;
    then ex X9 being Pythagorean_triple st X = X9 & X9 is non degenerate & X9
    is simplified;
    hence X is finite;
  end;
  hence thesis by A4,FINSET_1:7;
end;
