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 Th54:
  (not x in L) & L is being_line implies ex x1,x2 st L = Line(x1,
  x2) & x - x1 _|_ x2 - x1
proof
  assume that
A1: not x in L and
A2: L is being_line;
  consider y0,y1 such that
A3: y0 <> y1 and
A4: L = Line(y0,y1) by A2;
A5: y0 - y1 <> 0*n by A3,Th9;
  consider x1 being Element of REAL n such that
A6: x1 in Line(y0,y1) and
A7: y0 - y1,x - x1 are_orthogonal by Th43;
  x - x1 <> 0*n by A1,A4,A6,Th9;
  then
A8: y0 - y1 _|_ x - x1 by A7,A5;
  take x1;
  consider x2 being Element of REAL n such that
A9: x1 <> x2 and
A10: x2 in L by A2,EUCLID_4:14;
  take x2;
A11: x2 - x1 <> 0*n by A9,Th9;
  ex a being Real st x2 - x1 = a*(y0 - y1) by A4,A6,A10,Th31;
  then x2 - x1 // y0 - y1 by A5,A11;
  hence thesis by A2,A4,A6,A8,A9,A10,Th30,Th44;
end;
