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 p1,p2,p3 st p2-p1,p3-p1 are_lindependent2 holds inside_of_triangle
  (p1,p2,p3) is non empty
proof
  let p1,p2,p3;
  assume
A1: p2-p1,p3-p1 are_lindependent2;
  set p0=(1/3)*p1+(1/3)*p2+(1/3)*p3;
  set i01=tricord1(p1,p2,p3,p0),i02=tricord2(p1,p2,p3,p0), i03=tricord3(p1,p2,
  p3,p0);
  p0 in the carrier of TOP-REAL 2;
  then p0 in REAL 2 by EUCLID:22;
  then
A2: (1/3)+(1/3)+(1/3)=1 & p0 in plane(p1,p2,p3) by A1,Th54;
  then
A3: 1/3=i03 by A1,Def13;
  1/3=i01 & 1/3=i02 by A1,A2,Def11,Def12;
  hence thesis by A1,A3,Th58;
end;
