
theorem Th02:
  for a being non zero Real, b, r being Real
      st r = sqrt (a^2 + b^2) holds
  r is positive & (a / r)^2 + (b / r)^2 = 1
  proof
    let a be non zero Real;
    let b,r be Real;
    assume
A1: r = sqrt (a^2 + b^2);
    thus r is positive by A1,SQUARE_1:25;
    reconsider r as positive Real by A1,SQUARE_1:25;
    (a / r)^2 = a^2 / r^2 & (b / r)^2 = b^2 / r^2 by XCMPLX_1:76;
    then (a / r)^2 + (b / r)^2 = (a^2 + b^2) / r^2
                              .= r^2 / r^2 by A1,SQUARE_1:def 2
                              .= 1 by XCMPLX_1:60;
    hence thesis;
  end;
