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
  x in v + Lin{w} iff ex a st x = v + a * w
proof
  thus x in v + Lin{w} implies ex a st x = v + a * w
  proof
    assume x in v + Lin{w};
    then consider u being VECTOR of V such that
A1: u in Lin{w} and
A2: x = v + u by RLSUB_1:63;
    ex a st u = a * w by A1,Th8;
    hence thesis by A2;
  end;
  given a such that
A3: x = v + a * w;
  a * w in Lin{w} by Th8;
  hence thesis by A3,RLSUB_1:63;
end;
