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 Th16:
  for X,Y be ComplexLinearSpace, f,h be VECTOR of
C_VectorSpace_of_LinearOperators(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 ComplexLinearSpace;
  let f,h be VECTOR of C_VectorSpace_of_LinearOperators(X,Y);
  reconsider f9=f,h9=h as LinearOperator of X,Y by Def4;
  let c be Complex;
A1: C_VectorSpace_of_LinearOperators(X,Y) is Subspace of ComplexVectSpace(
  the carrier of X,Y) by Th13,CSSPACE:11;
  then reconsider f1=f as VECTOR of ComplexVectSpace(the carrier of X,Y) by
CLVECT_1:29;
  reconsider h1=h as VECTOR of ComplexVectSpace(the carrier of X,Y) by A1,
CLVECT_1:29;
A2: now
    assume
A3: h = c*f;
    let x be Element of X;
    h1 = c*f1 by A1,A3,CLVECT_1:33;
    hence h9.x=c*f9.x by Th12;
  end;
  now
    assume for x be Element of X holds h9.x=c*f9.x;
    then h1=c*f1 by Th12;
    hence h =c*f by A1,CLVECT_1:33;
  end;
  hence thesis by A2;
end;
