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 Th79:
  for L1,L2 st L1 is being_line & L2 is being_line & L1 <> L2
  holds ex x st x in L1 & not x in L2
proof
  let L1,L2;
  assume that
A1: L1 is being_line and
A2: L2 is being_line;
  consider x1,x2 such that
A3: x1 <> x2 and
A4: L1 = Line(x1,x2) by A1;
  assume
A5: L1 <> L2;
  now
    per cases by A2,A3,A4,A5,Th30;
    case
A6:   not x1 in L2;
      set x = x1;
      thus x in L1 & not x in L2 by A4,A6,EUCLID_4:9;
    end;
    case
A7:   not x2 in L2;
      set x = x2;
      thus x in L1 & not x in L2 by A4,A7,EUCLID_4:9;
    end;
  end;
  hence thesis;
end;
