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 (AS is right_transitive iff for u,u1,v,
v1,u2,v2 holds ( u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2<>v2 implies u,u1 // v,
  v1) )
proof
A1: (for u,u1,u2,v,v1,v2,w,w1 being Element of AS holds ((u,u1 '//' v,v1 & v
,v1 '//' w,w1 & u2,v2 '//' w,w1 ) implies (w=w1 or v=v1 or u,u1 '//' u2,v2 )))
implies for u,u1,v,v1,u2,v2 holds ( u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2<>v2
  implies u,u1 // v,v1)
  proof
    set w = the Element of AS;
    assume
A2: for u,u1,u2,v,v1,v2,w,w1 being Element of AS holds u,u1 '//' v,v1
& v,v1 '//' w,w1 & u2,v2 '//' w,w1 implies (w=w1 or v=v1 or u,u1 '//' u2,v2 );
    let u,u1,v,v1,u2,v2;
    assume that
A3: u,u1 '//' u2,v2 and
A4: v,v1 '//' u2,v2 and
A5: u2<>v2;
    consider w1 such that
A6: w<>w1 and
A7: w,w1 '//' u,u1 by Def1;
A8: now
      set w = the Element of AS;
      assume
A9:   u=u1;
      consider w1 such that
A10:  w<>w1 and
A11:  w,w1 '//' v,v1 by Def1;
      w,w1 '//' u,u by Def1;
      hence thesis by A9,A10,A11;
    end;
    now
      assume u<>u1;
      then w,w1 '//' v,v1 by A2,A3,A4,A5,A7;
      hence thesis by A6,A7;
    end;
    hence thesis by A8;
  end;
  assume
A12: AS is bach_transitive;
  (for u,u1,v,v1,u2,v2 holds ((u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2<>v2
  ) implies u,u1 // v,v1)) implies for u,u1,u2,v,v1,v2,w,w1 being Element of AS
holds ( u,u1 '//' v,v1 & v,v1 '//' w,w1 & u2,v2 '//' w,w1 implies (w=w1 or v=v1
  or u,u1 '//' u2,v2 ))
  proof
    assume
A13: for u,u1,v,v1,u2,v2 holds u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2
    <>v2 implies u,u1 // v,v1;
    let u,u1,u2,v,v1,v2,w,w1;
    assume that
A14: u,u1 '//' v,v1 and
A15: v,v1 '//' w,w1 and
A16: u2,v2 '//' w,w1;
    now
      assume that
A17:  w<>w1 and
      v<>v1;
      v,v1 // u2,v2 by A13,A15,A16,A17;
      then ex u3,v3 st u3<>v3 & u3,v3 '//' v,v1 & u3,v3 '//' u2,v2;
      hence thesis by A12,A14;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
