theorem
  a<>b & a,b,c are_collinear & a,b '||' c,d implies c,b '||' c,d
proof
  assume that
A1: a<>b and
A2: a,b,c are_collinear and
A3: a,b '||' c,d;
  now
    a,b '||' a,c by A2;
    then
A4: a,c '||' c,b & a,c '||' c,d by A1,A3,PARSP_1:24,def 12;
    assume a<>c;
    hence thesis by A4,PARSP_1:def 12;
  end;
  hence thesis by A3;
end;
