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 Th26:
for RS be RealLinearSpace,
      f be FinSequence of RS,
      x be set holds
   x in Z_Lin(f) iff
     ex g be (len f)-element FinSequence of RS,
          a be (len f)-element INT-valued FinSequence st
          x=Sum(g) & for i be Nat st i in Seg (len f) holds
          g/.i=(a.i)*(f/.i)
proof
let RS be RealLinearSpace,
      f be FinSequence of RS,
      x be set;
 hereby assume x in Z_Lin(f); then
consider g be (len f)-element FinSequence of RS such that
A1: x=Sum(g) & ex s be (len f)-element INT-valued FinSequence
          st for i be Nat st i in Seg (len f) holds
          g/.i=(s.i)*(f/.i);
consider s be (len f)-element INT-valued FinSequence such that
A2: for i be Nat st i in Seg (len f) holds g/.i=(s.i)*(f/.i) by A1;
take g,s;
thus x=Sum(g) & for i be Nat st i in Seg (len f) holds
          g/.i=(s.i)*(f/.i) by A1,A2;
end;
assume ex g be (len f)-element FinSequence of RS,
          a be (len f)-element INT-valued FinSequence st
          x=Sum(g) & for i be Nat st i in Seg (len f) holds
          g/.i=(a.i)*(f/.i);
hence x in Z_Lin(f);
end;
