 reserve n for Nat;

theorem THQQ:
  OASpace TOP-REAL 2 is OAffinSpace
  proof
    (ex u,v be VECTOR of TOP-REAL 2 st
      for a,b being Real st a*u + b*v = 0.(TOP-REAL 2) holds a=0 & b=0)
    proof
      reconsider u = |[1,0]|, v = |[0,1]| as VECTOR of TOP-REAL 2;
      now
        let a,b be Real;
        assume
A1:     a * u + b * v = 0.(TOP-REAL 2);
A2:     a * u + b * v = |[a * 1,a * 0]| + b * v by EUCLID:58
                     .= |[a * 1,a * 0]| + |[b * 0, b * 1]| by EUCLID:58
                     .= |[a + 0,0 + b]| by EUCLID:56
                     .= |[a,b]|;
        |[a,b]|`1 = a & |[a,b]|`2 = b & |[0,0]|`1 = 0 & |[0,0]|`2 = 0
          by EUCLID:52;
        hence a = 0 & b = 0 by A1,A2,EUCLID:54;
      end;
      hence thesis;
    end;
    hence thesis by ANALOAF:26;
  end;
