theorem
  a<>b & a,b,c are_collinear & a,b '||' c,d implies a,c '||' b,d
proof
  assume that
A1: a<>b and
A2: a,b,c are_collinear and
A3: a,b '||' c,d;
  now
A4: a,b '||' a,c by A2;
    then a,b '||' c,b by PARSP_1:24;
    then c,b '||' c,d by A1,A3,PARSP_1:def 12;
    then
A5: b,c '||' b,d by PARSP_1:24;
    assume
A6: b<>c;
    b,c '||' a,c by A4,PARSP_1:24;
    hence thesis by A6,A5,PARSP_1:def 12;
  end;
  hence thesis by A3;
end;
