let AS be Oriented_Orthogonality_Space; :: thesis: ( AS is bach_transitive implies ( AS is right_transitive iff for u, u1, v, v1, u2, v2 being Element of AS st u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 holds
u,u1 // v,v1 ) )
A1: ( ( for u, u1, u2, v, v1, v2, w, w1 being Element of AS st u,u1 '//' v,v1 & v,v1 '//' w,w1 & u2,v2 '//' w,w1 & not w = w1 & not v = v1 holds
u,u1 '//' u2,v2 ) implies for u, u1, v, v1, u2, v2 being Element of AS st u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 holds
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 st u,u1 '//' v,v1 & v,v1 '//' w,w1 & u2,v2 '//' w,w1 & not w = w1 & not v = v1 holds
u,u1 '//' u2,v2 ; :: thesis: for u, u1, v, v1, u2, v2 being Element of AS st u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 holds
u,u1 // v,v1
let u, u1, v, v1, u2, v2 be Element of AS; :: thesis: ( u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 implies u,u1 // v,v1 )
assume that
A3: u,u1 '//' u2,v2 and
A4: v,v1 '//' u2,v2 and
A5: u2 <> v2 ; :: thesis: u,u1 // v,v1
consider w1 being Element of AS such that
A6: the Element of AS <> w1 and
A7: the Element of AS,w1 '//' u,u1 by Def1;

A8: now :: thesis: ( u = u1 implies u,u1 // v,v1 )

set w = the Element of AS;
assume A9: u = u1 ; :: thesis: u,u1 // v,v1
consider w1 being Element of AS such that
A10: the Element of AS <> w1 and
A11: the Element of AS,w1 '//' v,v1 by Def1;
the Element of AS,w1 '//' u,u by Def1;
hence u,u1 // v,v1 by A9, A10, A11; :: thesis: verum

end;

now :: thesis: ( u <> u1 implies u,u1 // v,v1 )

assume u <> u1 ; :: thesis: u,u1 // v,v1
then the Element of AS,w1 '//' v,v1 by A2, A3, A4, A5, A7;
hence u,u1 // v,v1 by A6, A7; :: thesis: verum

end;

hence u,u1 // v,v1 by A8; :: thesis: verum

end;

assume A12: AS is bach_transitive ; :: thesis: ( AS is right_transitive iff for u, u1, v, v1, u2, v2 being Element of AS st u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 holds
u,u1 // v,v1 )
( ( for u, u1, v, v1, u2, v2 being Element of AS st u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 holds
u,u1 // v,v1 ) implies for u, u1, u2, v, v1, v2, w, w1 being Element of AS st u,u1 '//' v,v1 & v,v1 '//' w,w1 & u2,v2 '//' w,w1 & not w = w1 & not v = v1 holds
u,u1 '//' u2,v2 )

proof

assume A13: for u, u1, v, v1, u2, v2 being Element of AS st u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 holds
u,u1 // v,v1 ; :: thesis: for u, u1, u2, v, v1, v2, w, w1 being Element of AS st u,u1 '//' v,v1 & v,v1 '//' w,w1 & u2,v2 '//' w,w1 & not w = w1 & not v = v1 holds
u,u1 '//' u2,v2
let u, u1, u2, v, v1, v2, w, w1 be Element of AS; :: thesis: ( u,u1 '//' v,v1 & v,v1 '//' w,w1 & u2,v2 '//' w,w1 & not w = w1 & not v = v1 implies u,u1 '//' u2,v2 )
assume that
A14: u,u1 '//' v,v1 and
A15: v,v1 '//' w,w1 and
A16: u2,v2 '//' w,w1 ; :: thesis: ( w = w1 or v = v1 or u,u1 '//' u2,v2 )

now :: thesis: ( w <> w1 & v <> v1 & not w = w1 & not v = v1 implies u,u1 '//' u2,v2 )

assume that
A17: w <> w1 and
v <> v1 ; :: thesis: ( w = w1 or v = v1 or u,u1 '//' u2,v2 )
v,v1 // u2,v2 by A13, A15, A16, A17;
then ex u3, v3 being Element of AS st
( u3 <> v3 & u3,v3 '//' v,v1 & u3,v3 '//' u2,v2 ) ;
hence ( w = w1 or v = v1 or u,u1 '//' u2,v2 ) by A12, A14; :: thesis: verum

end;

hence ( w = w1 or v = v1 or u,u1 '//' u2,v2 ) ; :: thesis: verum

end;

hence ( AS is right_transitive iff for u, u1, v, v1, u2, v2 being Element of AS st u,u1 '//' u2,v2 & v,v1 '//' u2,v2 & u2 <> v2 holds
u,u1 // v,v1 ) by A1; :: thesis: verum