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 Th62:
  (not x in L) & L is being_line implies ex L0 st x in L0 & L0 _|_
  L & L0 meets L
proof
  assume ( not x in L)& L is being_line;
  then consider x1,x2 such that
A1: L = Line(x1,x2) and
A2: (x - x1) _|_ (x2 - x1) by Th54;
  reconsider L0 = Line(x1,x) as Subset of REAL n;
  reconsider L0 as Element of line_of_REAL n by Th47;
  x1 in L0 & x1 in L by A1,EUCLID_4:9;
  then
A3: x in L0 & L0 meets L by Th49,EUCLID_4:9;
  L0 _|_ L by A1,A2;
  hence thesis by A3;
end;
