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;
reserve L,L0,L1,L2 for Element of line_of_REAL n;

theorem
  x2-x1,x3-x1 are_lindependent2 & a1 + a2 + a3 = 1 & x = a1*x1+a2*x2+a3*
x3 & b1 + b2 + b3 = 1 & x = b1*x1+b2*x2+b3*x3 implies a1 = b1 & a2 = b2 & a3 =
  b3
proof
  assume
A1: x2-x1,x3-x1 are_lindependent2;
  assume that
A2: a1 + a2 + a3 = 1 and
A3: x = a1*x1+a2*x2+a3*x3 and
A4: b1 + b2 + b3 = 1 and
A5: x = b1*x1+b2*x2+b3*x3;
  a1 = 1 - a2 -a3 by A2;
  then x = 1 * x1 - a2*x1 - a3*x1 + a2*x2 + a3*x3 by A3,Th13
    .= x1 + -a2*x1 - a3*x1 + a2*x2 + a3*x3 by EUCLID_4:3
    .= x1 + -a2*x1 + a2*x2 + -a3*x1 + a3*x3 by RVSUM_1:15
    .= x1 + (a2*x2 + -a2*x1) + -a3*x1 + a3*x3 by RVSUM_1:15
    .= x1 + (a2*x2 + -a2*x1) + (a3*x3 + -a3*x1) by RVSUM_1:15
    .= x1 + (a2*x2 + a2*(-x1)) + (a3*x3 + -a3*x1) by Th3
    .= x1 + (a2*x2 + a2*(-x1)) + (a3*x3 + a3*(-x1)) by Th3
    .= x1 + a2*(x2 + -x1) + (a3*x3 + a3*(-x1)) by EUCLID_4:6
    .= x1 + a2*(x2 - x1) + a3*(x3 - x1) by EUCLID_4:6;
  then
A6: x - x1 = a2*(x2 - x1) + a3*(x3 - x1) by Th7;
  x = (1 - b2 -b3)*x1 + b2*x2 + b3*x3 by A4,A5
    .= 1 * x1 - b2*x1 - b3*x1 + b2*x2 + b3*x3 by Th13
    .= x1 + -b2*x1 - b3*x1 + b2*x2 + b3*x3 by EUCLID_4:3
    .= x1 + -b2*x1 + b2*x2 + -b3*x1 + b3*x3 by RVSUM_1:15
    .= x1 + (b2*x2 + -b2*x1) + -b3*x1 + b3*x3 by RVSUM_1:15
    .= x1 + (b2*x2 + -b2*x1) + (b3*x3 + -b3*x1) by RVSUM_1:15
    .= x1 + (b2*x2 + b2*(-x1)) + (b3*x3 + -b3*x1) by Th3
    .= x1 + (b2*x2 + b2*(-x1)) + (b3*x3 + b3*(-x1)) by Th3
    .= x1 + b2*(x2 + -x1) + (b3*x3 + b3*(-x1)) by EUCLID_4:6
    .= x1 + b2*(x2 - x1) + b3*(x3 - x1) by EUCLID_4:6;
  then a2*(x2 - x1) + a3*(x3 - x1) = b2*(x2 - x1) + b3*(x3 - x1) by A6,Th7;
  then a2 = b2 & a3 = b3 by A1,Th35;
  hence thesis by A2,A4;
end;
