reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

theorem Th15:
  ex x st not x in L
  proof
    assume
A1: x in L;
    reconsider OO = |[0,0]|, Ox = |[1,0]|, Oy = |[0,1]| as Element of REAL 2
    by EUCLID:22;
A2: OO in L & Ox in L & Oy in L by A1;
    per cases by Th6;
    suppose L is being_line;
      then L = Line(Ox,Oy) by A2,Th14,EUCLIDLP:30;
      then OO in the set of all (1-lambda) * Ox+lambda * Oy by A1;
      then consider lambda such that
A3:   OO = (1-lambda) * Ox + lambda * Oy;
A4:   |[0,0]| = (1-lambda) * |[1,0]| + lambda * |[0,1]| by A3
             .= |[(1-lambda) * 1 , (1-lambda) * 0]| + lambda * |[0,1]|
                   by EUCLID:58
             .= |[1-lambda , 0]| + |[lambda * 0,lambda * 1]| by EUCLID:58
             .= |[1-lambda + 0 , 0 + lambda]| by EUCLID:56
             .= |[1-lambda,lambda]|;
      |[0,0]|`1 = 0 & |[0,0]|`2 = 0 & |[1-lambda,lambda]|`1=1-lambda &
      |[1-lambda,lambda]|`2 = lambda by EUCLID:52;
      hence contradiction by A4;
    end;
    suppose ex x st L={x};
      then consider x such that
A5:   L = {x};
      Ox in {x} & Oy in {x} by A5,A1;
      then Ox = x & Oy = x by TARSKI:def 1;
      hence contradiction by Th14;
    end;
  end;
