
theorem Th44:
  for V be VectSp of F_Complex, v,w be Vector of V for f be
  diagReR+0valued hermitan-Form of V st signnorm(f,w)= 0 holds |.f.(w,v).| = 0
proof
  let V be VectSp of F_Complex, v1,w be Vector of V, f be diagReR+0valued
  hermitan-Form of V;
  set w1 = f.(w,v1), A = signnorm(f,v1), B = |.w1.|, C = signnorm(f,w);
  reconsider A as Real;
  assume that
A1: C = 0 and
A2: B <> 0;
A3: ex a be Element of F_Complex st |.a.| =1 & Re (a * w1) = |.w1.| & Im (a *
  w1)= 0 by Th8;
A4: now
A5: now
      assume
A6:   A >0;
      A - 2*B * (A/B) + C*(A/B)^2 = A - (A*(2*B))/B by A1,XCMPLX_1:74
        .= A - (A*2)*B/B
        .= A - (A+A) by A2,XCMPLX_1:89
        .= - A;
      hence contradiction by A3,A6,Th43;
    end;
A7: now
      assume
A8:   A <0;
      0 <= A - 2*B*0 + C * 0^2 by A3,Th43;
      hence contradiction by A8;
    end;
    assume A<>0;
    hence contradiction by A7,A5;
  end;
  now
    assume
A9: A= 0;
A10: now
      assume
A11:  0 < B;
      0 <= A - 2*B*1 + C * 1^2 by A3,Th43;
      hence contradiction by A1,A9,A11;
    end;
    now
      assume
A12:  B < 0;
      0 <= A - 2*B*(-1) + C * (-1)^2 by A3,Th43;
      hence contradiction by A1,A9,A12;
    end;
    hence contradiction by A2,A10;
  end;
  hence contradiction by A4;
end;
