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 Th64:
  x1 in L & x2 in L & x1 <> x2 implies Line(x1,x2) = L & L is being_line
proof
  assume that
A1: x1 in L & x2 in L and
A2: x1 <> x2;
A3: Line(x1,x2) c= L by A1,Th48;
  L in line_of_REAL n;
  then ex y1,y2 being Element of REAL n st L = Line(y1,y2);
  then L c= Line(x1,x2) by A1,A2,EUCLID_4:11;
  then Line(x1,x2) = L by A3,XBOOLE_0:def 10;
  hence thesis by A2;
end;
