reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;

theorem Th3:
  x,y // z,t implies x,y // t,z & y,x // z,t & y,x // t,z &
  z,t // x,y & z,t // y,x & t,z // x,y & t,z // y,x
proof
  assume
A1: x,y // z,t;
  hence x,y // t,z & y,x // z,t by Lm2,Lm3;
  hence y,x // t,z by Lm2;
  thus z,t // x,y by A1,Lm1;
  hence z,t // y,x & t,z // x,y by Lm2,Lm3;
  hence thesis by Lm3;
end;
