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 Th60:
  for p being Prime st p = 4*k+1
   ex a,b being positive Nat st a > b & p = a^2+b^2
  proof
    let p be Prime;
    assume
A1: p = 4*k+1;
    then p mod 4 = 1 by Lm6,NAT_D:21;
    then consider a,b such that
A2: p = a^2+b^2 by NAT_5:23;
    now
      assume 0 >= a;
      then a = 0;
      hence contradiction by A2;
    end;
    then reconsider a as positive Nat;
    now
      assume 0 >= b;
      then b = 0;
      hence contradiction by A2;
    end;
    then reconsider b as positive Nat;
    a <> b
    proof
      assume a = b;
      then
A3:   p = 2*(a^2) by A2;
      p = 2*(2*k)+1 by A1;
      hence contradiction by A3;
    end;
    then per cases by XXREAL_0:1;
    suppose
A4:   a < b;
      take b,a;
      thus thesis by A2,A4;
    end;
    suppose
A5:   a > b;
      take a,b;
      thus thesis by A2,A5;
    end;
  end;
