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 Th76:
  for x1,x2,L st L is being_line & L = Line(x1,x2) holds x1 <> x2
proof
  let x1,x2,L;
  assume that
A1: L is being_line and
A2: L = Line(x1,x2);
  consider y1, y2 such that
A3: y1 <> y2 and
A4: L = Line(y1,y2) by A1;
  y1 in L & y2 in L by A4,EUCLID_4:9;
  then consider a such that
A5: y2 - y1 = a*(x2 - x1) by A2,Th31;
  thus x1 <> x2
  proof
    assume x1 = x2;
    then y2 - y1 = a* 0*n by A5,Th2
      .= 0*n by EUCLID_4:2;
    hence contradiction by A3,Th9;
  end;
end;
