
theorem Th2:
  for X,Y,Z be ComplexNormSpace for f be Lipschitzian LinearOperator of
X,Y for g be Lipschitzian LinearOperator of Y,Z holds
g*f is Lipschitzian LinearOperator
  of X,Z & for x be VECTOR of X holds ||.((g*f).x).|| <=(
  BoundedLinearOperatorsNorm(Y,Z).g) *(BoundedLinearOperatorsNorm(X,Y).f )*||.x
.|| & (BoundedLinearOperatorsNorm(X,Z).(g*f)) <=(BoundedLinearOperatorsNorm(Y,Z
  ).g) *(BoundedLinearOperatorsNorm(X,Y).f)
proof
  let X,Y,Z be ComplexNormSpace;
  let f be Lipschitzian LinearOperator of X,Y;
  let g be Lipschitzian LinearOperator of Y,Z;
  reconsider ff=f as Point of C_NormSpace_of_BoundedLinearOperators(X,Y) by
CLOPBAN1:def 7;
  reconsider gg=g as Point of C_NormSpace_of_BoundedLinearOperators(Y,Z) by
CLOPBAN1:def 7;
A1: now
    let v be VECTOR of X;
    0 <= ||.gg.|| by CLVECT_1:105;
    then
A2: ||.gg.||*||.f.(v).|| <=||.gg.||*(||.ff.||*||.v.||) by CLOPBAN1:31
,XREAL_1:64;
    ||.(g*f).v.|| = ||.g.(f.v).|| & ||.g.(f.(v)).|| <=||.gg.|| * ||.f.(v)
    .|| by CLOPBAN1:31,FUNCT_2:15;
    hence ||.(g*f).v.|| <=(||.gg.|| * ||.ff.||) * ||.v.|| by A2,XXREAL_0:2;
  end;
  set K = ||.gg.|| * ||.ff.||;
A3: 0 <= ||.gg.|| & 0 <= ||.ff.|| by CLVECT_1:105;
  then reconsider gf=g*f as Lipschitzian LinearOperator of X,Z by A1,Th1,
CLOPBAN1:def 6;
A4: now
    let t be VECTOR of X;
    assume ||.t.|| <= 1;
    then
A5: K*||.t.|| <=K*1 by A3,XREAL_1:64;
    ||.(g*f).t.|| <=K* ||.t.|| by A1;
    hence ||.(g*f).t.|| <=K by A5,XXREAL_0:2;
  end;
A6: now
    let r be Real;
    assume r in PreNorms(gf);
    then ex t be VECTOR of X st r=||.gf.t.|| & ||.t.|| <= 1;
    hence r <=K by A4;
  end;
  (for s be Real st s in PreNorms(gf) holds s <=K) implies upper_bound
  PreNorms(gf) <=K by SEQ_4:45;
  hence thesis by A1,A6,CLOPBAN1:29;
end;
