
theorem Th45:
  for V be VectSp of F_Complex, v,w be Vector of V for f be
  diagReR+0valued hermitan-Form of V holds |. f.(v,w) .|^2 <= signnorm(f,v) *
  signnorm(f,w)
proof
  let V be VectSp of F_Complex, v1,w be Vector of V, f be diagReR+0valued
  hermitan-Form of V;
  set v4=f.(v1,w), w1=f.(w,v1), A = signnorm(f,v1), B = |.w1.|, C = signnorm(f
  ,w);
  reconsider A,B,C as Real;
A1: C = 0^2 implies B^2 <=A*C by Th44;
A2: ex a be Element of F_Complex st |.a.| =1 & Re (a * w1) = |.w1.| & Im (a *
  w1)= 0 by Th8;
A3: C > 0 implies B^2 <=A*C
  proof
    assume
A4: C > 0;
    A - 2*B *(B/C) + C*(B/C)^2 = A - (B*(2*B)/C) + C*(B/C)^2 by XCMPLX_1:74
      .= A - (B^2*2)/C + C*(B/C)^2
      .= A - 2 *(B^2/C) + C*(B/C)^2 by XCMPLX_1:74
      .= A - 2 *(B^2/C) + C*(B^2/(C*C)) by XCMPLX_1:76
      .= A - 2 *(B^2/C) + (C*B^2)/(C*C) by XCMPLX_1:74
      .= A - 2 *(B^2/C) + B^2/C by A4,XCMPLX_1:91
      .= A - B^2/C
      .= (A*C - B^2)/C by A4,XCMPLX_1:127;
    then 0 <= A*C - B^2 by A2,A4,Th43;
    then 0+ B^2 <= A*C - B^2 + B^2 by XREAL_1:6;
    hence thesis;
  end;
  B= |.v4*'.| by Def5
    .= |.v4.| by COMPLEX1:53;
  hence thesis by A1,A3,Def7;
end;
