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 Th6:
  rng l c= INT implies rng (i*l) c= INT
proof
  assume A1:rng l c= INT;
    let y be object;
    assume A2:y in rng (i*l);
    consider x be object such that
A3: x in the carrier of V & y=(i*l).x by A2,FUNCT_2:11;
    reconsider z=x as VECTOR of V by A3;
    reconsider ii=i as Real;
    l.z in rng l by FUNCT_2:4; then
    reconsider z1=l.z as Integer by A1;
    (i*l).z = (ii*l).z by Th4
           .= i*z1 by RLVECT_2:def 11;
    hence y in INT by A3,INT_1:def 2;
end;
