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 Th101:
  x in P & L c= P & (not x in L) & L is being_line implies P is being_plane
proof
  assume
A1: x in P & L c= P;
  assume ( not x in L)& L is being_line;
  then consider x1,x2 being Element of REAL n such that
A2: L = Line(x1,x2) and
A3: x - x1,x2 - x1 are_lindependent2 by Th55;
  x1 in L & x2 in L by A2,EUCLID_4:9;
  then P = plane(x1,x,x2) by A1,A3,Th92;
  hence thesis by A3;
end;
