
theorem Th2:
  for n being Nat, p1, p2 being Point of TOP-REAL n, r1,
r2 being Real st (1-r1)*p1+r1*p2 = (1-r2)*p1+r2*p2 holds r1 = r2 or p1 =
  p2
proof
  let n be Nat, p1, p2 be Point of TOP-REAL n, r1, r2 be Real;
  assume
A1: (1-r1)*p1+r1*p2 = (1-r2)*p1+r2*p2;
A2: 1*p1+(-r1*p1+r1*p2) = 1*p1+-r1*p1+r1*p2 by RLVECT_1:def 3
    .= 1*p1-r1*p1+r1*p2
    .= (1-r2)*p1+r2*p2 by A1,RLVECT_1:35
    .= 1*p1-r2*p1+r2*p2 by RLVECT_1:35
    .= 1*p1+-r2*p1+r2*p2
    .= 1*p1+(-r2*p1+r2*p2) by RLVECT_1:def 3;
A3: (r2-r1)*p1+r1*p2 = r2*p1-r1*p1+r1*p2 by RLVECT_1:35
    .= r2*p1+-r1*p1+r1*p2
    .= r2*p1+(-r1*p1+r1*p2) by RLVECT_1:def 3
    .= r2*p1+(-r2*p1+r2*p2) by A2,GOBOARD7:4
    .= r2*p1+-r2*p1+r2*p2 by RLVECT_1:def 3
    .= 0.TOP-REAL n + r2*p2 by RLVECT_1:5
    .= r2*p2 by RLVECT_1:4;
  (r2-r1)*p1 = (r2-r1)*p1 + 0.TOP-REAL n by RLVECT_1:4
    .= (r2-r1)*p1+(r1*p2 - r1*p2) by RLVECT_1:5
    .= r2*p2 - r1*p2 by A3,RLVECT_1:def 3
    .= (r2-r1)*p2 by RLVECT_1:35;
  then r2-r1=0 or p1 = p2 by RLVECT_1:36;
  hence thesis;
end;
