reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th39:
  for Bn being Subset of RealVectSpace(Seg n), x,y being Element
of REAL n,a being Real st Bn is linearly-independent & x in Bn & y in Bn & y=a*
  x holds x=y
proof
  let Bn be Subset of RealVectSpace(Seg n), x,y be Element of REAL n,a be Real;
  assume that
A1: Bn is linearly-independent and
A2: x in Bn and
A3: y in Bn and
A4: y=a*x;
  reconsider x0=x,y0=y as Element of RealVectSpace(Seg n) by FINSEQ_2:93;
  reconsider A={y0,x0} as Subset of RealVectSpace(Seg n);
  A c= Bn
  by A2,A3,TARSKI:def 2;
  then
A5: A is linearly-independent by A1,RLVECT_3:5;
  reconsider aa=a as Element of REAL by XREAL_0:def 1;
  y0=aa*x0 by A4;
  hence x=y by A5,RLVECT_3:12;
end;
