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{w1,w2} iff ex a,b st x = v + a * w1 + b * w2
proof
  thus x in v + Lin{w1,w2} implies ex a,b st x = v + a * w1 + b * w2
  proof
    assume x in v + Lin{w1,w2};
    then consider u such that
A1: u in Lin{w1,w2} and
A2: x = v + u by RLSUB_1:63;
    consider a,b such that
A3: u = a * w1 + b * w2 by A1,Th11;
    take a,b;
    thus thesis by A2,A3,RLVECT_1:def 3;
  end;
  given a,b such that
A4: x = v + a * w1 + b * w2;
  a * w1 + b * w2 in Lin{w1,w2} by Th11;
  then v + (a * w1 + b * w2) in v + Lin{w1,w2} by RLSUB_1:63;
  hence thesis by A4,RLVECT_1:def 3;
end;
