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 Th24:
  for X,Y be ComplexNormSpace, f,h be VECTOR of
  C_VectorSpace_of_BoundedLinearOperators(X,Y), c be Complex holds h = c*f iff
  for x be VECTOR of X holds h.x = c * f.x
proof
  let X,Y be ComplexNormSpace;
  let f,h be VECTOR of C_VectorSpace_of_BoundedLinearOperators(X,Y);
  let c be Complex;
A1: C_VectorSpace_of_BoundedLinearOperators(X,Y) is Subspace of
  C_VectorSpace_of_LinearOperators(X,Y) by Th21,CSSPACE:11;
  then reconsider f1=f as VECTOR of C_VectorSpace_of_LinearOperators(X,Y) by
CLVECT_1:29;
  reconsider h1=h as VECTOR of C_VectorSpace_of_LinearOperators(X,Y) by A1,
CLVECT_1:29;
  hereby
    assume
A2: h = c*f;
    let x be Element of X;
    h1=c*f1 by A1,A2,CLVECT_1:33;
    hence h.x=c*f.x by Th16;
  end;
  assume for x be Element of X holds h.x=c*f.x;
  then h1=c*f1 by Th16;
  hence thesis by A1,CLVECT_1:33;
end;
