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 Th35:
  for x1,x2 st x1,x2 are_lindependent2 holds a1*x1+a2*x2=b1*x1+b2*
  x2 implies a1=b1 & a2=b2
proof
  let x1,x2;
  assume
A1: x1,x2 are_lindependent2;
  assume
A2: a1*x1+a2*x2=b1*x1+b2*x2;
  0*n = (a1*x1+a2*x2) - (a1*x1+a2*x2) by Th2
    .= (a1-b1)*x1 + (a2-b2)*x2 by A2,Th25;
  then a1 - b1 = 0 & a2 - b2 = 0 by A1;
  hence thesis;
end;
