
theorem Th40:
  for CLSP being CollSp
  for p,q being Element of CLSP holds Line(p,q) = Line(q,p)
  proof
    let CLSP be CollSp;
    let p,q be Element of CLSP;
A1: Line(p,q) c= Line(q,p)
    proof
      let x be object;
      assume x in Line(p,q);
      then x in {y where y is Element of CLSP: p,q,y are_collinear}
        by COLLSP:def 5;
      then consider y be Element of CLSP such that
A2:   y = x and
A3:   p,q,y are_collinear;
      q,p,y are_collinear by A3,COLLSP:4;
      then y in {y where y is Element of CLSP: q,p,y are_collinear};
      hence thesis by A2,COLLSP:def 5;
    end;
    Line(q,p) c= Line(p,q)
    proof
      let x be object;
      assume x in Line(q,p);
      then x in {y where y is Element of CLSP: q,p,y are_collinear}
        by COLLSP:def 5;
      then consider y be Element of CLSP such that
A4:  y = x and
A5:  q,p,y are_collinear;
      p,q,y are_collinear by A5,COLLSP:4;
      then y in {y where y is Element of CLSP: p,q,y are_collinear};
      hence thesis by A4,COLLSP:def 5;
    end;
    hence thesis by A1;
  end;
