
theorem Th03:
  for       a being non zero Real,
      b,c,d,e being Real st
  a * b = c - d * e holds
  b^2 = (c^2/a^2) - 2 * ((c * d) / (a * a)) * e + (d^2 / a^2) * e^2
  proof
    let a be non zero Real;
    let b,c,d,e be Real;
    assume
A1: a * b = c - d * e;
    b = (a * b) / a by XCMPLX_1:89
     .= (c / a - (d * e) / a) by A1,XCMPLX_1:120
     .= (c / a - (d / a) * e);
    then b^2 = (c / a)^2 - 2 * (c / a) * ((d / a) * e) + ((d / a) * e)^2
            .= (c^2 / a^2) - 2 * (c / a) * ((d / a) * e) + ((d / a)^2 * e^2)
              by XCMPLX_1:76
            .= (c^2 / a^2) - 2 * ((c / a) * (d / a)) * e + ((d^2 / a^2) * e^2)
              by XCMPLX_1:76
            .= (c^2 / a^2) - 2 * ((c * d) / (a * a)) * e + ((d^2 / a^2) * e^2)
              by XCMPLX_1:76;
    hence thesis;
  end;
