reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;
reserve V for RealLinearSpace,
        p,q,x for Element of V;
reserve p, q, x for Point of TOP-REAL n;
reserve s, t for Point of TOP-REAL 2;

theorem
  (a*s + b*t)`1 = a * s`1 + b * t`1
proof
  thus (a*s+b*t)`1 = (a*s)`1 + (b*t)`1 by TOPREAL3:2
    .= a*s`1 + (b*t)`1 by TOPREAL3:4
    .= a*s`1+b*t`1 by TOPREAL3:4;
end;
