theorem
  (ex p,q st p <> q) implies (Collinear a,b,c iff
  (ex A st A is_line & a in A & b in A & c in A))
  proof
    assume ex p,q st p <> q;
    then consider p,q such that
A1: p <> q;
A2: (ex A st A is_line & a in A & b in A & c in A) implies Collinear a,b,c
    proof
      assume ex A st A is_line & a in A & b in A & c in A;
      then consider A such that
A3:   A is_line and
A4:   a in A and
A5:   b in A and
A6:   c in A;
      per cases;
      suppose a <> b;
        then c in Line(a,b) by A3,A4,A5,A6,Satz6p18;
        then ex x being POINT of S st c = x & Collinear a,b,x;
        hence thesis;
      end;
      suppose a = b;
        hence thesis by Satz3p1;
      end;
    end;
    Collinear a,b,c implies (ex A st A is_line & a in A & b in A & c in A)
    proof
      assume
A8:   Collinear a,b,c;
      per cases;
      suppose
A9:     a = b;
        per cases;
        suppose
A10:      a = c;
          per cases by A1;
          suppose
A11:        a <> p;
            set A = Line(a,p);
            now
              thus A is_line by A11;
              Collinear a,p,a by Satz3p1;
              hence a in A & b in A & c in A by A9,A10;
            end;
            hence ex A st A is_line & a in A & b in A & c in A;
          end;
          suppose
A12:        a <> q;
            set A = Line(a,q);
            now
              thus A is_line by A12;
              Collinear a,q,a by Satz3p1;
              hence a in A & b in A & c in A by A9,A10;
            end;
            hence ex A st A is_line & a in A & b in A & c in A;
          end;
        end;
        suppose
A13:      a <> c;
          set A = Line(a,c);
          now
            thus A is_line by A13;
            Collinear a,c,a & Collinear a,c,c by Satz3p1;
            hence a in A & b in A & c in A by A9;
          end;
          hence ex A st A is_line & a in A & b in A & c in A;
        end;
      end;
      suppose
A14:    a <> b;
        Collinear a,b,a & Collinear a,b,b by Satz3p1;
        then
A15:    a in Line(a,b) & b in Line(a,b) & c in Line(a,b) by A8;
        reconsider A = Line(a,b) as Subset of S;
        A is_line by A14;
        hence ex A st A is_line & a in A & b in A & c in A by A15;
      end;
    end;
    hence thesis by A2;
  end;
