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 Th102:
  (not x in L) & L is being_line & x in P0 & L c= P0 & x in P1 &
  L c= P1 implies P0 = P1
proof
  assume that
A1: ( not x in L)& L is being_line and
A2: x in P0 & L c= P0 and
A3: x in P1 & L c= P1;
  consider x1,x2 being Element of REAL n such that
A4: L = Line(x1,x2) and
A5: x - x1,x2 - x1 are_lindependent2 by A1,Th55;
A6: x1 in L & x2 in L by A4,EUCLID_4:9;
  then P0 = plane(x1,x,x2) by A2,A5,Th92;
  hence thesis by A3,A5,A6,Th92;
end;
