reserve X for non empty set;
reserve Y for ComplexLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Complex;
reserve u,v,w for VECTOR of CLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th20:
  for X,Y be ComplexNormSpace holds for f be LinearOperator of X,Y
  st (for x be VECTOR of X holds f.x=0.Y) holds f is Lipschitzian
proof
  let X,Y be ComplexNormSpace;
  let f be LinearOperator of X,Y such that
A1: for x be VECTOR of X holds f.x=0.Y;
A2: now
    let x be VECTOR of X;
    thus ||. f.x .|| = ||. 0.Y .|| by A1
      .=0*||. x .|| by CLVECT_1:102;
  end;
  take 0;
  thus thesis by A2;
end;
