
theorem
  for V being RealUnitarySpace, v,w1,w2 being VECTOR of V, x being set
  holds x in v + Lin{w1,w2} iff ex a,b being Real st x = v + a * w1 + b * w2
proof
  let V be RealUnitarySpace;
  let v,w1,w2 be VECTOR of V;
  let x be set;
  thus x in v + Lin{w1,w2} implies ex a,b being Real st x = v + a * w1 + b * w2
  proof
    assume x in v + Lin{w1,w2};
    then consider u being VECTOR of V such that
A1: u in Lin{w1,w2} and
A2: x = v + u by RUSUB_2:63;
    consider a,b being Real such that
A3: u = a * w1 + b * w2 by A1,Th32;
    take a,b;
    thus thesis by A2,A3,RLVECT_1:def 3;
  end;
  given a,b be Real such that
A4: x = v + a * w1 + b * w2;
  a * w1 + b * w2 in Lin{w1,w2} by Th32;
  then v + (a * w1 + b * w2) in v + Lin{w1,w2} by RUSUB_2:63;
  hence thesis by A4,RLVECT_1:def 3;
end;
