reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;
reserve e,e1,e2 for Element of C_LinComb V;

theorem Th39:
  a * vector(LC_CLSpace V,L) = a * L
proof
  L in the carrier of LC_CLSpace V by Def12;
  then L in LC_CLSpace V;
  then
A1: (C_LCMult V).[a,vector(LC_CLSpace V,L)] = (C_LCMult V).[a,@L] by
RLVECT_2:def 1;
  @@L = L;
  hence thesis by A1,Def16;
end;
