theorem Th31:
  a,b // A & a,b // p,q & a<>b implies p,q // A
proof
  assume that
A1: a,b // A and
A2: a,b // p,q and
A3: a<>b;
A4: A is being_line by A1;
  then consider c,d such that
A5: c <>d and
A6: c in A and
A7: d in A and
A8: a,b // c,d by A1,Th29;
  p,q // c,d by A2,A3,A8,Th4;
  hence thesis by A4,A5,A6,A7,Th29;
end;
