let V be RealLinearSpace; :: thesis: for x, y, u, v, u1, v1, w being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,u1,w are_COrtm_wrt x,y & not u,v,v1,w are_COrtm_wrt x,y holds
u,v,w,v1 are_COrtm_wrt x,y
let x, y, u, v, u1, v1, w be VECTOR of V; :: thesis: ( Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,u1,w are_COrtm_wrt x,y & not u,v,v1,w are_COrtm_wrt x,y implies u,v,w,v1 are_COrtm_wrt x,y )
assume A1:
( Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,u1,w are_COrtm_wrt x,y )
; :: thesis: ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y )
then A2:
( Ortm x,y,u, Ortm x,y,v // u1,v1 & Ortm x,y,u, Ortm x,y,v // u1,w )
by Def4;
A3:
now assume A4:
(
u <> v &
u1 <> v1 )
;
:: thesis: ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y )then
u1,
v1 // u1,
w
by A1, A2, Th6, ANALOAF:20;
then A5:
(
u1,
v1 // v1,
w or
u1,
w // w,
v1 )
by ANALOAF:23;
now assume A6:
u1 <> w
;
:: thesis: ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y )
(
u1,
v1 // Ortm x,
y,
u,
Ortm x,
y,
v &
u1,
w // Ortm x,
y,
u,
Ortm x,
y,
v )
by A2, ANALOAF:21;
then
(
Ortm x,
y,
u,
Ortm x,
y,
v // v1,
w or
Ortm x,
y,
u,
Ortm x,
y,
v // w,
v1 )
by A4, A5, A6, ANALOAF:20;
hence
(
u,
v,
v1,
w are_COrtm_wrt x,
y or
u,
v,
w,
v1 are_COrtm_wrt x,
y )
by Def4;
:: thesis: verum end; hence
(
u,
v,
v1,
w are_COrtm_wrt x,
y or
u,
v,
w,
v1 are_COrtm_wrt x,
y )
by A1;
:: thesis: verum end;
now assume
u = v
;
:: thesis: ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y )then
(
Ortm x,
y,
u,
Ortm x,
y,
v // v1,
w or
Ortm x,
y,
u,
Ortm x,
y,
v // w,
v1 )
by ANALOAF:18;
hence
(
u,
v,
v1,
w are_COrtm_wrt x,
y or
u,
v,
w,
v1 are_COrtm_wrt x,
y )
by Def4;
:: thesis: verum end;
hence
( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y )
by A1, A3; :: thesis: verum