reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem
  for p being Prime st p = 4*k+1
   ex a,b being positive Nat st p^2 = a^2+b^2
  proof
    let p be Prime;
    assume p = 4*k+1;
    then consider a,b being positive Nat such that
A1: a > b and
A2: p = a^2+b^2 by Th60;
A3: a-b > b-b by A1,XREAL_1:9;
    then reconsider c = a-b as Nat by INT_1:3;
    c > 0 by A3;
    then reconsider c as positive Nat;
    a^2-b^2 = c*(a+b);
    then reconsider x = a^2-b^2 as positive Nat;
    reconsider y = 2*a*b as positive Nat;
    take x,y;
    thus thesis by A2;
  end;
