reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;

theorem Th25:
  s = (1-r)*p + r*q & s <> p & 0 <= r implies 0 < r
proof
  assume that
A1: s = (1-r)*p + r*q and
A2: s <> p and
A3: 0 <= r;
  assume
A4: r <= 0;
  then s = (1-0)*p + r*q by A1,A3
    .= (1-0)*p + 0 * q by A3,A4
    .= 1 * p + 0.TOP-REAL 2 by RLVECT_1:10
    .= 1 * p by RLVECT_1:4
    .= p by RLVECT_1:def 8;
  hence thesis by A2;
end;
