reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th44:
  L is Linear_Combination of A implies a * L is Linear_Combination of A
proof
  assume
A1: L is Linear_Combination of A;
  now
    per cases;
    suppose
      a = 0;
      then a * L = ZeroLC(V) by Th43;
      hence thesis by Th22;
    end;
    suppose
      a <> 0;
      then Carrier(a * L) = Carrier(L) by Th42;
      hence thesis by A1,Def6;
    end;
  end;
  hence thesis;
end;
