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 Th5:
  rng l1 c= INT & rng l2 c= INT implies rng (l1+l2) c= INT
proof
  assume A1: rng l1 c= INT & rng l2 c= INT;
    let y be object;
    assume A2:y in rng (l1+l2);
    consider x be object such that
A3: x in the carrier of V & y=(l1+l2).x by A2,FUNCT_2:11;
    reconsider z=x as VECTOR of V by A3;
    l1.z in rng l1 by FUNCT_2:4; then
    reconsider z1=l1.z as Integer by A1;
    l2.z in rng l2 by FUNCT_2:4; then
    reconsider z2=l2.z as Integer by A1;
    (l1+l2).z = z1+ z2 by RLVECT_2:def 10;
    hence y in INT by A3,INT_1:def 2;
end;
