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;
reserve P,P0,P1,P2 for Element of plane_of_REAL n;

theorem
  L1 is being_line & L2 is being_line & L1 <> L2 & L1 c= P1 & L2 c= P1 &
  L1 c= P2 & L2 c= P2 implies P1 = P2
proof
  assume that
A1: L1 is being_line and
A2: L2 is being_line and
A3: L1 <> L2 and
A4: L1 c= P1 & L2 c= P1 and
A5: L1 c= P2 & L2 c= P2;
  consider x such that
A6: x in L1 and
A7: not x in L2 by A1,A2,A3,Th79;
  consider x1,x2 such that
A8: L2 = Line(x1,x2) and
A9: x - x1,x2 - x1 are_lindependent2 by A2,A7,Th55;
A10: x1 in L2 & x2 in L2 by A8,EUCLID_4:9;
  then P2 = plane(x1,x,x2) by A5,A6,A9,Th92;
  hence thesis by A4,A6,A9,A10,Th92;
end;
