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 Th12:
  for X be non empty set, Y be ComplexLinearSpace, f,h be VECTOR
of ComplexVectSpace(X,Y), c be Complex holds h = c*f iff for x be Element of X
  holds h.x = c * f.x
proof
  let X be non empty set;
  let Y be ComplexLinearSpace;
  let f,h be VECTOR of ComplexVectSpace(X,Y);
  let c be Complex;
  reconsider f9=f,h9=h as Element of Funcs(X, the carrier of Y);
  hereby
    assume
A1: h = c*f;
    let x be Element of X;
    h= (FuncExtMult(X, Y)).[c,f9] by A1,CLVECT_1:def 1;
    hence h.x = c*f.x by Th2;
  end;
  assume for x be Element of X holds h.x=c*f.x;
  then h9=(FuncExtMult(X, Y)).[c,f9] by Th2;
  hence thesis by CLVECT_1:def 1;
end;
