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

theorem Th39:
  not ex q,s,t being Prime st q^2 = s^2 + t^2
  proof
    given q,s,t being Prime such that
A1: q^2 = s^2 + t^2;
    per cases by A1,Th38;
    suppose that
A2:   s is even and t is odd;
      s = 2 by A2,LAGRA4SQ:13;
      then q^2 - t^2 = 4 by A1;
      hence thesis by Th35;
    end;
    suppose that s is odd and
A3:   t is even;
      t = 2 by A3,LAGRA4SQ:13;
      then q^2 - s^2 = 4 by A1;
      hence thesis by Th35;
    end;
  end;
