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 Th35:
  for X,Y be ComplexNormSpace, f,h be Point of
C_NormSpace_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 Point of C_NormSpace_of_BoundedLinearOperators(X,Y);
  let c be Complex;
  reconsider f1=f as VECTOR of C_VectorSpace_of_BoundedLinearOperators(X,Y);
  reconsider h1=h as VECTOR of C_VectorSpace_of_BoundedLinearOperators(X,Y);
A1: now
    assume h1=c*f1;
    hence
    h=Mult_(BoundedLinearOperators(X,Y), C_VectorSpace_of_LinearOperators
    (X,Y)).[c,f1] by CLVECT_1:def 1
      .=c*f by CLVECT_1:def 1;
  end;
  now
    assume h=c*f;
    hence h1=Mult_(BoundedLinearOperators(X,Y),
    C_VectorSpace_of_LinearOperators(X,Y)).[c,f] by CLVECT_1:def 1
      .=c*f1 by CLVECT_1:def 1;
  end;
  hence thesis by A1,Th24;
end;
