
theorem Th47:
  for V be VectSp of F_Complex for f be diagReR+0valued
  hermitan-Form of V for v,w be Vector of V holds signnorm(f,v+w) <= (sqrt(
  signnorm(f,v)) + sqrt(signnorm(f,w)))^2
proof
  let V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V, v,w
  be Vector of V;
  set v3 = f.(v,v), v4 = f.(v,w), w1 = f.(w,v), w2 = f.(w,w), sv = signnorm(f,
  v), sw = signnorm(f,w), svw = signnorm(f,v+w);
A1: 0 <= sv by Def7;
A2: svw = Re (v3 + v4 +(w1+w2)) by BILINEAR:28
    .= Re (v3 + v4)+ Re(w1+w2) by HAHNBAN1:10
    .= Re v3 + Re v4 + Re(w1+w2) by HAHNBAN1:10
    .= Re v3 + Re v4 + (Re w1 + Re w2) by HAHNBAN1:10
    .= sv + (Re v4 + Re w1) + sw
    .= sv + (Re v4 + Re (v4*')) + sw by Def5
    .= sv + 2*(Re v4) + sw by Th15
    .= sv + sw+ 2*(Re v4);
A3: 0 <= sw by Def7;
  0 <= |.v4.| by COMPLEX1:46;
  then sqrt(|.v4.|^2) <= sqrt(sv * sw) by Th45,SQUARE_1:26;
  then |.v4.| <= sqrt(sv * sw) by COMPLEX1:46,SQUARE_1:22;
  then Re v4 <= |.v4.| & |.v4.| <= sqrt(sv) * sqrt(sw) by A1,A3,COMPLEX1:54
,SQUARE_1:29;
  then Re v4 <= sqrt(sv) * sqrt(sw) by XXREAL_0:2;
  then 2* (Re v4) <= 2*(sqrt(sv) * sqrt(sw)) by XREAL_1:64;
  then svw <= sv + sw + 2*(sqrt(sv) * sqrt(sw)) by A2,XREAL_1:6;
  then svw <= (sqrt sv)^2 + sw + 2*(sqrt(sv) * sqrt(sw)) by A1,SQUARE_1:def 2;
  then svw <= (sqrt sv)^2 + (sqrt sw)^2 + 2*(sqrt(sv) * sqrt(sw)) by A3,
SQUARE_1:def 2;
  hence thesis;
end;
