theorem Satz6p19:
  a <> b & A is_line & a in A & b in A & B is_line & a in B & b in B
  implies A = B
  proof
    assume that
A1: a <> b and
A2: A is_line and
A3: a in A and
A4: b in A and
A5: B is_line and
A6: a in B and
A7: b in B;
    consider pa,qa be POINT of S such that pa <> qa and
A8: A = Line(pa,qa) by A2;
    consider pb,qb be POINT of S such that pb <> qb and
A9: B = Line(pb,qb) by A5;
    Line(pa,qa) = Line(a,b) & Line(pb,qb) = Line(a,b)
      by A1,A3,A4,A8,A2,A6,A7,A9,A5,Satz6p18;
    hence thesis by A8,A9;
  end;
