reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem Th25:
  for x,y,z being Element of TOP-REAL 1 holds x,y,z are_collinear
  proof
    let x,y,z be Element of TOP-REAL 1;
    per cases;
    suppose
A1:   z <> y;
      reconsider L = Line(y,z) as line of TOP-REAL 1;
      H1: y in L & z in L by RLTOPSP1:72;
      reconsider x1 = x,y1 = y,z1 = z as Element of REAL 1 by EUCLID:22;
A2:   x1 in REAL 1 & y1 in REAL 1 & z1 in REAL 1;
      consider xr be Element of REAL such that
A3:   x = <* xr *> by A2,FINSEQ_2:97;
      consider yr be Element of REAL such that
A4:   y = <* yr *> by A2,FINSEQ_2:97;
      consider zr be Element of REAL such that
A5:   z = <* zr *> by A2,FINSEQ_2:97;
A6:   zr - yr <> 0 by A4,A5,A1;
      reconsider r = (xr - yr) / (zr - yr) as Real;
A7:   (1 - r) * yr + r * zr = yr + (xr -yr) / (zr - yr) * (zr - yr)
                           .= yr + (xr - yr) by A6,XCMPLX_1:87
                           .= xr;
      (1 - r) * y1 + r * z1 = <* (1 - r) * yr *> + r * <* zr *>
                                  by A4,A5,RVSUM_1:47
                           .= <* (1 - r) * yr *> + <* r * zr *> by RVSUM_1:47
                           .= x by RVSUM_1:13,A3,A7;
      then x in the set of all (1 - r) * y1 + r * z1 where r is Real;
      then x in Line(y1,z1) by EUCLID_4:def 1;
      then x in L by EUCLID12:4;
      hence thesis by H1;
    end;
    suppose z = y;
      then x in Line(x,y) & y in Line(x,y) & z in Line(x,y) by RLTOPSP1:72;
      hence thesis;
    end;
  end;
