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
  x1,x2 are_lindependent2 implies x1+t*x2,x2 are_lindependent2
proof
  assume
A1: x1,x2 are_lindependent2;
  for a,b being Real st a*(x1+t*x2)+b*x2=0*n holds a=0 & b=0
  proof
    let a,b;
    assume a*(x1+t*x2)+b*x2=0*n;
    then
A2: 0*n = a*(1 *x1+t*x2)+b*x2 by EUCLID_4:3
      .= (a*1)*x1+(a*t)*x2+b*x2 by Th21
      .= a*x1+((a*t)*x2+b*x2) by RVSUM_1:15
      .= a*x1+(a*t+b)*x2 by EUCLID_4:7;
    then a= 0 by A1;
    hence thesis by A1,A2;
  end;
  hence thesis;
end;
