reserve x,y for set;
reserve a,b for Real;
reserve i,j for Integer;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
reserve v,v1,v2,v3,u,w,w1,w2,w3 for VECTOR of V;
reserve A,B,C for Subset of V;
reserve L,L1,L2 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem
  for RS be RealLinearSpace,
      f be non empty FinSequence of RS holds
    Z_Lin (rng f) = Z_Lin f
proof
let RS be RealLinearSpace,
    f be non empty FinSequence of RS;
 for x be object holds x in Z_Lin (rng f) iff x in Z_Lin f
proof
  let x be object;
  hereby assume x in Z_Lin (rng f); then
    consider g,h be FinSequence of RS,
        a be INT-valued FinSequence such that
A1:x=Sum(h) & rng g c= rng f & len g =len h & len g = len a &
     for i be Nat st i in Seg (len g) holds
     h/.i=(a.i)*(g/.i) by Lm2;
  rng f c= Z_Lin f by Th31;
  hence x in Z_Lin f by A1,Th32,XBOOLE_1:1;
end;
assume x in Z_Lin f;
then consider g be (len f)-element FinSequence of RS,
       a be (len f)-element INT-valued FinSequence such that
  A2:  x=Sum(g) & for i be Nat st i in Seg (len f) holds
          g/.i=(a.i)*(f/.i) by Th26;
  len f =len g & len a = len f by CARD_1:def 7;
  hence x in Z_Lin (rng f) by A2,Lm1;
end;
hence thesis by TARSKI:2;
end;
