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 iff for u,u1,u2,v,v1,v2 holds (u,u1 '//' v
  ,v1 & v,v1 // u2,v2 & v<>v1 implies u,u1 '//' u2,v2)
proof
A1: (for u,u1,u2,v,v1,v2 holds (u,u1 '//' v,v1 & v,v1 // u2,v2 & v<>v1
implies u,u1 '//' u2,v2)) implies for u,u1,u2,v,v1,v2,w,w1 being Element of AS
holds ( u,u1 '//' v,v1 & w,w1 '//' v,v1 & w,w1 '//' u2,v2 implies (w=w1 or v=v1
  or u,u1 '//' u2,v2) )
  proof
    assume
A2: for u,u1,u2,v,v1,v2 holds (u,u1 '//' v,v1 & v,v1 // u2,v2 & v<>v1
    implies u,u1 '//' u2,v2);
    let u,u1,u2,v,v1,v2,w,w1;
    assume that
A3: u,u1 '//' v,v1 and
A4: w,w1 '//' v,v1 and
A5: w,w1 '//' u2,v2;
    now
      assume that
A6:   w<>w1 and
      v<>v1;
      v,v1 // u2,v2 by A4,A5,A6;
      hence thesis by A2,A3;
    end;
    hence thesis;
  end;
  thus thesis by A1;
end;
