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 Th81:
  for x,L1,L2 st x in L2 & L1 _|_ L2 holds ex x0 st x <> x0 & x0
  in L1 & not x0 in L2
proof
  let x,L1,L2;
  assume that
A1: x in L2 and
A2: L1 _|_ L2;
  L1 is being_line & L2 is being_line by A2,Th67;
  then ex x0 st x0 in L1 & not x0 in L2 by A2,Th75,Th79;
  hence thesis by A1;
end;
