let AS be Oriented_Orthogonality_Space; ( AS is bach_transitive implies for u, u1, v, v1, w, w1 being Element of AS st u,u1 // v,v1 & v,v1 // w,w1 & v <> v1 holds
u,u1 // w,w1 )
assume A1:
AS is bach_transitive
; for u, u1, v, v1, w, w1 being Element of AS st u,u1 // v,v1 & v,v1 // w,w1 & v <> v1 holds
u,u1 // w,w1
let u, u1, v, v1, w, w1 be Element of AS; ( u,u1 // v,v1 & v,v1 // w,w1 & v <> v1 implies u,u1 // w,w1 )
assume that
A2:
u,u1 // v,v1
and
A3:
v,v1 // w,w1
and
A4:
v <> v1
; u,u1 // w,w1
consider v2, v3 being Element of AS 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
u,u1 // w,w1
by A5, A6; verum