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