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 Th7:
  rng (ZeroLC(V)) c= INT
  proof
    let y be object;
    assume A1:y in rng (ZeroLC(V));
    consider x be object such that
A2: x in the carrier of V & y=(ZeroLC(V)).x by A1,FUNCT_2:11;
    reconsider z = x as VECTOR of V by A2;
    ZeroLC(V).z = 0 by RLVECT_2:20;
    hence y in INT by A2,NUMBERS:17;
  end;
