
theorem Th4:
  for V being RealLinearSpace, OAS being OAffinSpace st OAS=OASpace
(V) holds for a,b,c,d being Element of OAS,u,v,w,y being VECTOR of V st a=u & b
  =v & c =w & d=y holds (a,b '||' c,d iff u,v '||' w,y)
proof
  let V be RealLinearSpace, OAS be OAffinSpace such that
A1: OAS=OASpace(V);
  let a,b,c,d be Element of OAS,u,v,w,y be VECTOR of V;
  assume
A2: a=u & b=v & c =w & d=y;
A3: now
    assume u,v '||' w,y;
    then u,v // w,y or u,v // y,w by GEOMTRAP:def 1;
    then a,b // c,d or a,b // d,c by A1,A2,GEOMTRAP:2;
    hence a,b '||' c,d by DIRAF:def 4;
  end;
  now
    assume a,b '||' c,d;
    then a,b // c,d or a,b // d,c by DIRAF:def 4;
    then u,v // w,y or u,v // y,w by A1,A2,GEOMTRAP:2;
    hence u,v '||' w,y by GEOMTRAP:def 1;
  end;
  hence thesis by A3;
end;
