theorem
  p <> q & p <> r & between q,p,r implies
  Line(p,q) = halfline(p,q) \/ {p} \/ halfline(p,r)
  proof
    assume that
A1A: p <> q and
A1B: p <> r and
A1C: between q,p,r;
    thus Line(p,q) c= halfline(p,q) \/ {p} \/ halfline(p,r)
    proof
      let t be object;
      assume t in Line(p,q);
      then consider x be Element of S such that
A2:   t = x & Collinear p,q,x;
A3:   between q,x,p & p <> x implies x in halfline(p,q)
      proof
        assume between q,x,p & p <> x;
        then p out x,q by A1A,Satz3p2;
        hence thesis;
      end;
A4:   between x,p,q & x = q implies p = x by GTARSKI1:def 10;
A5:   between x,p,q & p <> x & x <> q implies x in halfline(p,r)
      proof
        assume
A6:     between x,p,q & p <> x & x <> q;
        then between q,p,r & between q,p,x by A1C,Satz3p2;
        then p out x,r by A1A,A1B,A6,Satz5p2;
        hence thesis;
      end;
      between p,q,x & p <> x implies x in halfline(p,q)
      proof
        assume between p,q,x & p <> x;
        then p out x,q by A1A;
        hence thesis;
      end;
      then (x in halfline(p,q) or x in {p}) or x in halfline(p,r)
        by A2,A3,A5,A4,TARSKI:def 1;
      then (x in halfline(p,q) \/ {p}) or x in halfline(p,r) by XBOOLE_0:def 3;
      hence thesis by A2,XBOOLE_0:def 3;
    end;
    thus halfline(p,q) \/ {p} \/ halfline(p,r) c= Line(p,q)
    proof
      let t be object;
      assume t in halfline(p,q) \/ {p} \/ halfline(p,r);
      then t in halfline(p,q) \/ {p} or t in halfline(p,r) by XBOOLE_0:def 3;
      then t in halfline(p,q) or t in {p} or t in halfline(p,r)
        by XBOOLE_0:def 3;
      then per cases by TARSKI:def 1;
      suppose t in halfline(p,q);
        then consider x be POINT of S such that
A7:     t = x & p out x,q;
        Collinear p,q,x by A7,Satz3p2;
        hence thesis by A7;
      end;
      suppose
A8:     t = p;
        then reconsider x = t as POINT of S;
        Collinear p,q,p by Satz3p1;
        hence thesis by A8;
      end;
      suppose t in halfline(p,r);
        then consider x be POINT of S such that
A9:     t = x & p out x,r;
A10:    between p,x,r implies between p,q,x or between q,x,p or between x,p,q
        proof
          assume between p,x,r;
          then between q,p,x by A1C,Satz3p5p1;
          hence thesis by Satz3p2;
        end;
        between p,r,x implies between p,q,x or between q,x,p or between x,p,q
        proof
          assume between p,r,x;
          then between q,p,x by A1B,A1C,Satz3p7p2;
          hence thesis by Satz3p2;
        end;
        then Collinear p,q,x by A9,A10;
         hence thesis by A9;
      end;
    end;
  end;
