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,w3} iff ex a,b,c st x = v + a * w1 + b * w2 + c * w3
proof
  thus x in v + Lin{w1,w2,w3} implies ex a,b,c st x = v + a * w1 + b * w2 + c
  * w3
  proof
    assume x in v + Lin{w1,w2,w3};
    then consider u such that
A1: u in Lin{w1,w2,w3} and
A2: x = v + u by RLSUB_1:63;
    consider a,b,c such that
A3: u = a * w1 + b * w2 + c * w3 by A1,Th14;
    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 such that
A4: x = v + a * w1 + b * w2 + c * w3;
  a * w1 + b * w2 + c * w3 in Lin{w1,w2,w3} by Th14;
  then v + (a * w1 + b * w2 + c * w3) in v + Lin{w1,w2,w3} by RLSUB_1:63;
  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;
