reserve z,z1,z2 for Complex;
reserve r,x1,x2 for Real;
reserve p0,p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem
  for n being Element of NAT,p1,p2,p3 being Point of TOP-REAL n holds
plane(p1,p2,p3)={p where p is Point of TOP-REAL n:
  ex a1,a2,a3 being Real st a1
  +a2+a3=1 & p=a1*p1+a2*p2+a3*p3}
proof
  let n be Element of NAT,p1,p2,p3 be Point of TOP-REAL n;
  thus plane(p1,p2,p3) c= {p where p is Point of TOP-REAL n: ex a1,a2,a3 being
  Real st a1+a2+a3=1 & p=a1*p1+a2*p2+a3*p3}
  proof
    let x be object;
    assume
A1: x in plane(p1,p2,p3);
    then reconsider p0=x as Point of TOP-REAL n;
    ex a1,a2,a3 being Real
st a1+a2+a3=1 & p0=a1*p1+a2*p2+a3 *p3 by A1,Th48;
    hence thesis;
  end;
  let x be object;
  assume x in {p where p is Point of TOP-REAL n:
   ex a1,a2,a3 being Real st a1
  +a2+a3=1 & p=a1*p1+a2*p2+a3*p3};
  then
  ex p being Point of TOP-REAL n st p=x &
ex a1,a2,a3 being Real st a1+a2+
  a3=1 & p=a1*p1+a2*p2+a3*p3;
  hence thesis by Th52;
end;
