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 Th12:
  for OO,Ox,Oy be Element of REAL 2 st
  OO = |[0,0]| & Ox = |[1,0]| & Oy = |[0,1]| holds
  REAL 2 = plane(OO,Ox,Oy)
  proof
    let OO,Ox,Oy be Element of REAL 2 such that
A1: OO = |[0,0]| and
A2: Ox = |[1,0]| and
A3: Oy = |[0,1]|;
    now
      let a be object;
      assume a in REAL 2;
      then reconsider b = a as Point of TOP-REAL 2 by EUCLID:22;
A4:   b = |[b`1 + 0 , 0 + b`2]| by EUCLID:53
       .= |[ b`1 * 1 , b`1 * 0]| + |[ b`2 * 0 , b`2 *1 ]| by EUCLID:56
       .= b`1 * |[1,0]| + |[ b`2 * 0 , b`2 *1 ]| by EUCLID:58
       .= b`1 * |[1,0]| + b`2 * |[0,1]| by EUCLID:58;
      reconsider a1 = 1 - (b`1+b`2) as Real;
      reconsider a2 = b`1 as Real;
      reconsider a3 = b`2 as Real;
      a1 * |[0,0]| = |[a1 *0,a1 * 0]| by EUCLID:58
                  .= |[0,0]|; then
      a1 * |[0,0]| + b = |[0,0]| + |[b`1,b`2]| by EUCLID:53
                      .= |[0 + b`1,0 + b`2]| by EUCLID:56
                      .= b by EUCLID:53;
      then
A5:   b = (a1 * |[0,0]|) + (a2 * |[1,0]|) + (a3 * |[0,1]|)
            by A4,RLVECT_1:def 3;
      a1+a2+a3 = 1 & b = a1 * OO + a2 * Ox + a3 * Oy by A1,A2,A5,A3;
      then a in {x where x is Element of REAL 2 : ex a1,a2,a3 be Real st
      a1+a2+a3=1 & x = a1*OO+a2*Ox+a3*Oy};
      hence a in plane(OO,Ox,Oy) by EUCLIDLP:def 9;
    end;
    then REAL 2 c= plane(OO,Ox,Oy);
    hence thesis;
  end;
