reserve V for RealLinearSpace;
reserve x,y for VECTOR of V;
reserve AS for Oriented_Orthogonality_Space;
reserve u,u1,u2,u3,v,v1,v2,v3,w,w1 for Element of AS;

theorem
  AS is bach_transitive implies for u,u1,v,v1,w,w1 holds (u,u1 // v,v1 &
  v,v1 // w,w1 & v<>v1 implies u,u1 // w,w1)
proof
  assume
A1: AS is bach_transitive;
  let u,u1,v,v1,w,w1;
  assume that
A2: u,u1 // v,v1 and
A3: v,v1 // w,w1 and
A4: v<>v1;
  consider v2,v3 such that
A5: v2<>v3 and
A6: v2,v3 '//' u,u1 and
A7: v2,v3 '//' v,v1 by A2;
  v2,v3 '//' w,w1 by A1,A3,A4,A7;
  hence thesis by A5,A6;
end;
