reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;

theorem Th37:
  x1,x2 are_lindependent2 implies x1 <> x2
proof
  assume
A1: x1,x2 are_lindependent2;
  assume
A2: x1 = x2;
  1 * x1 + (-1) * x2 = 1 * (x1 - x2) by Th12
    .= 1 * 0*n by A2,Th2
    .= 0*n by EUCLID_4:2;
  hence contradiction by A1;
end;
