
theorem
  for V being RealUnitarySpace, v,w1,w2,w3 being VECTOR of V, x being
set holds x in v + Lin{w1,w2,w3} iff ex a,b,c being Real st x = v + a * w1 + b
  * w2 + c * w3
proof
  let V be RealUnitarySpace;
  let v,w1,w2,w3 be VECTOR of V;
  let x be set;
  thus x in v + Lin{w1,w2,w3} implies ex a,b,c being Real st x = v + a * w1 +
  b * w2 + c * w3
  proof
    assume x in v + Lin{w1,w2,w3};
    then consider u being VECTOR of V such that
A1: u in Lin{w1,w2,w3} and
A2: x = v + u by RUSUB_2:63;
    consider a,b,c being Real such that
A3: u = a * w1 + b * w2 + c * w3 by A1,Th35;
    take a,b,c;
    x = v + (a * w1 + b * w2) + c * w3 by A2,A3,RLVECT_1:def 3;
    hence thesis by RLVECT_1:def 3;
  end;
  given a,b,c be Real such that
A4: x = v + a * w1 + b * w2 + c * w3;
  a * w1 + b * w2 + c * w3 in Lin{w1,w2,w3} by Th35;
  then v + (a * w1 + b * w2 + c * w3) in v + Lin{w1,w2,w3} by RUSUB_2:63;
  then v + (a * w1 + b * w2) + c * w3 in v + Lin{w1,w2,w3} by RLVECT_1:def 3;
  hence thesis by A4,RLVECT_1:def 3;
end;
