reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;
reserve L,L0,L1,L2 for Element of line_of_REAL n;

theorem
  L0 // L1 & L1 // L2 implies L0 // L2
proof
  assume that
A1: L0 // L1 and
A2: L1 // L2;
  consider x0, x1, x2, x3 being Element of REAL n such that
A3: L0 = Line(x0,x1) and
A4: L1 = Line(x2,x3) and
A5: (x1 - x0) // (x3 - x2) by A1;
A6: x3 - x2 <> 0*n by A5;
  consider y0, y1, y2, y3 being Element of REAL n such that
A7: L1 = Line(y0,y1) and
A8: L2 = Line(y2,y3) and
A9: (y1 - y0) // (y3 - y2) by A2;
A10: y1 - y0 <> 0*n by A9;
  x3 in Line(y1,y0) & x2 in Line(y1,y0) by A4,A7,EUCLID_4:9;
  then ex a st x3 - x2 = a*(y1 - y0) by Th31;
  then (x3 - x2) // (y1 - y0) by A6,A10;
  then (x1 - x0) // (y1 - y0) by A5,Th33;
  then (x1 - x0) // (y3 - y2) by A9,Th33;
  hence thesis by A3,A8;
end;
