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 Th26:
  s = (1-r)*p + r*q & s <> q & r <= 1 implies r < 1
proof
  assume that
A1: s = (1-r)*p + r*q and
A2: s <> q and
A3: r <= 1;
  assume
A4: r >= 1;
  then s = (1-1)*p + r*q by A1,A3,XXREAL_0:1
    .= 0 * p + 1 * q by A3,A4,XXREAL_0:1
    .= 0.TOP-REAL 2 + 1 * q by RLVECT_1:10
    .= 1 * q by RLVECT_1:4
    .= q by RLVECT_1:def 8;
  hence thesis by A2;
end;
