reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  for a,b be non zero Nat, p be prime Nat st a + b = p holds a,b are_coprime
  proof
    let a,b be non zero Nat, p be prime Nat;
    A1: a+b > a+0 by XREAL_1:6;
    assume
    A2: a+b = p; then
    a,p are_coprime by A1,NAT_D:7,NAT_6:6;
    hence thesis by A2, NEWTON02:45;
  end;
