reserve K,F for Ring;
reserve V,W for VectSp of K;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;
reserve K for Ring;

theorem
  for G, F being non empty ModuleStr over K holds
  ( for x being set holds (x is Vector of [:G,F:]
  iff ex x1 being Vector of G, x2 being Vector of F st x=[x1,x2]) )
  & ( for x, y being Vector of [:G,F:],
  x1, y1 being Vector of G, x2, y2 being Vector of F
  st x = [x1,x2] & y = [y1,y2] holds x+y = [x1+y1,x2+y2] )
  & 0.[:G,F:] = [0.G,0.F]
  & ( for x being Vector of [:G,F:], x1 being Vector of G, x2 be Vector of F,
  a being Element of K
  st x = [x1,x2] holds a*x = [a*x1,a*x2] )
  by YDef2,PRVECT_3:def 1,SUBSET_1:43;
