reserve a,b,c,k,m,n for Nat;
reserve p for Prime;

theorem
  not 15,20 are_coprime & 15|^2+20|^2 = 5|^4 & not 7 divides 15*20
  proof
    (3*1+1) mod 3 = 1 by NUMBER02:16;
    then
A1: 1 = 3 gcd (4 mod 3) by NEWTON:51
    .= 4 gcd 3 by NAT_D:30;
    (5*3) gcd (5*4) = 5*(3 gcd 4) by PYTHTRIP:8;
    hence not 15,20 are_coprime by A1;
A2: 15|^2 = 15*15 & 20|^2 = 20*20 by WSIERP_1:1;
    5|^4 = 5*5*5*5 by POLYEQ_5:3;
    hence 15|^2+20|^2 = 5|^4 by A2;
    15*20 = 7*42+6;
    then 15*20 mod 7 = 6 by NUMBER02:16;
    hence not 7 divides 15*20 by PEPIN:6;
  end;
