theorem :: ExtPerp3:
  for a,b,c,d being POINT of S holds
    a <> b & b <> c & c <> d & a <> c & a <> d & b <> d &
      are_orthogonal b,a,a,c & Collinear a,c,d implies
        are_orthogonal b,a,a,d
  proof
    let a,b,c,d be POINT of S;
    assume
A1: a <> b & b <> c & c <> d & a <> c & a <> d & b <> d &
      are_orthogonal b,a,a,c & Collinear a,c,d; then
A3: d in Line(a,c) by LemmaA1;
    a in Line(a,c) by GTARSKI3:83;
    hence thesis by A1,A3,Prelim11;
  end;
