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
  x in v + Z_Lin{w} iff ex a be Integer st x = v + a * w
proof
  thus x in v + Z_Lin{w} implies ex a be Integer st x = v + a * w
  proof
    assume x in v + Z_Lin{w};
    then consider u being VECTOR of V such that
A1: x = v + u and
A2: u in Z_Lin{w};
    ex a be Integer st u = a * w by A2,Th16;
    hence thesis by A1;
  end;
  given a0 be Integer such that
A3: x = v + a0 * w;
  a0 * w in Z_Lin{w} by Th16;
  hence thesis by A3;
end;
