reserve x for set;
reserve a,b,c,d,e,r1,r2,r3,r4,r5,r6 for Real;
reserve V for RealLinearSpace;
reserve u,v,v1,v2,v3,w,w1,w2,w3 for VECTOR of V;
reserve W,W1,W2 for Subspace of V;

theorem
  W1 is Subspace of W2 implies v + W1 c= v + W2
proof
  assume
A1: W1 is Subspace of W2;
  let x be object;
  assume x in v + W1;
  then consider u such that
A2: u in W1 and
A3: x = v + u by RLSUB_1:63;
  u in W2 by A1,A2,RLSUB_1:8;
  hence thesis by A3,RLSUB_1:63;
end;
