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

theorem
  { 3,4,5 } is non degenerate simplified Pythagorean_triple
proof
  3^2 + 4^2 = 5^2;
  then reconsider X = { 3,4,5 } as Pythagorean_triple by Def4;
  3 gcd 4 = 3 gcd (4 - 3) by PREPOWER:97
    .= 1 by WSIERP_1:8;
  then
A1: 4 in X & 3,4 are_coprime by ENUMSET1:def 1;
  not 0 in X & 3 in X by ENUMSET1:def 1;
  hence thesis by A1,Def8,ORDINAL1:def 16;
end;
