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 Th31:
  for RS be RealLinearSpace,
      f be FinSequence of RS holds
    rng f c= Z_Lin(f)
proof
  let RS be RealLinearSpace, f be FinSequence of RS;
  let y be object;
    assume y in rng f; then
    consider x be object such that
A1: x in dom f & y=f.x by FUNCT_1:def 3;
A2: x in Seg len f by A1,FINSEQ_1:def 3;
    reconsider i=x as Nat by A1;
    y=f/.i by A1,PARTFUN1:def 6;
    hence y in Z_Lin(f) by A2,Th30;
end;
