
theorem Th09:
  for a     being non zero Real,
      b,c,d being Real st
  a^2 + c^2 = b^2 & 1 < b^2 holds
  not ((b^2)^2 / a^2) - 2 * ((b^2 * c) / (a * a)) * d
        + (c^2 / a^2) * d^2 + d^2 = 1
  proof
    let a be non zero Real;
    let b,c,d be Real;
    assume that
A1: a^2 + c^2 = b^2 and
A2: 1 < b^2;
    assume
A5: ((b^2)^2 / a^2) - 2 * ((b^2 * c) / (a * a)) * d
      + (c^2 / a^2) * d^2 + d^2 = 1;
    c^2 / a^2 + 1 = c^2 / a^2 + (a^2 / a^2) by XCMPLX_1:60
                 .= b^2 / a^2 by A1;
    then
A6: a^2 * (c^2 / a^2 + 1) = a^2 * b^2 / a^2 by XCMPLX_1:74
                         .= b^2 by XCMPLX_1:89;
A7: a^2 * (2 * ((b^2 * c) / (a * a))) = 2 * (a^2 * (b^2 * c) / (a * a))
                                     .= 2 * (b^2 * c) by XCMPLX_1:89;
A8: (a^2) * ((b^2)^2 / a^2) - (a^2) * 1
      = ((a^2) * (b^2)^2) / a^2 - (a^2) * 1
     .= (b^2)^2 - a^2 by XCMPLX_1:89;
    a^2 * 0 = a^2 * ((c^2 / a^2 + 1) * d^2 - 2 * ((b^2 * c) / (a * a)) * d
               + ((b^2)^2 / a^2) - 1) by A5;
    then
A9: 0 = (a^2) * (c^2 / a^2 + 1) * d^2
          - (a^2) * 2 * ((b^2 * c) / (a * a)) * d + (a^2) * ((b^2)^2 / a^2)
          - (a^2) * 1;
A11: b <> 0 by A6;
A12: 0 = (b^2 * d^2 - 2 * (b^2 * c) * d + (b^2)^2 - a^2)/(b^2) by A9,A6,A7,A8
      .= d^2 * b^2 / b^2 - 2 * (b^2 * c) * d / b^2
        + (b^2)^2 / b^2 - a^2 / b^2
      .= d^2 - 2 *  (b^2 * c) * d / b^2  + (b^2)^2 / b^2 - a^2 / b^2
        by A6,XCMPLX_1:89
      .= d^2 - 2 * c * d + (b^2)^2 / b^2 - a^2 / b^2 by A11,Th05
      .= d^2 + (-2 * c) * d + b^2 - a^2 / b^2 by A6,XCMPLX_1:89
      .= 1 * d^2 + (-2 * c) * d + (b^2 - a^2 / b^2);
    reconsider c1 = 1, c2 = -2 * c, c3 = b^2 - a^2 / b^2 as Real;
A13:a^2 / b^2 - a^2 < 0 by A2,Th08;
    delta(c1,c2,c3) < 0
    proof
      c2^2 - 4 * c1 * c3 = 4 * (c^2 - (b^2 - a^2 / b^2));
      hence thesis by A1,A13,QUIN_1:def 1;
    end;
    hence contradiction by A12,QUIN_1:3;
  end;
