let V be RealLinearSpace; for u, u1, u2, v, v1, v2, w, w1, x, y being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u2,v2,w,w1 are_COrtm_wrt x,y & not u,u1,u2,v2 are_COrtm_wrt x,y & not v = v1 holds
w = w1
let u, u1, u2, v, v1, v2, w, w1, x, y be VECTOR of V; ( Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u2,v2,w,w1 are_COrtm_wrt x,y & not u,u1,u2,v2 are_COrtm_wrt x,y & not v = v1 implies w = w1 )
assume A1:
Gen x,y
; ( not u,u1,v,v1 are_COrtm_wrt x,y or not v,v1,w,w1 are_COrtm_wrt x,y or not u2,v2,w,w1 are_COrtm_wrt x,y or u,u1,u2,v2 are_COrtm_wrt x,y or v = v1 or w = w1 )
assume that
A2:
u,u1,v,v1 are_COrtm_wrt x,y
and
A3:
v,v1,w,w1 are_COrtm_wrt x,y
and
A4:
u2,v2,w,w1 are_COrtm_wrt x,y
; ( u,u1,u2,v2 are_COrtm_wrt x,y or v = v1 or w = w1 )
Ortm (x,y,u), Ortm (x,y,u1) // v,v1
by A2;
then A5:
v,v1 // Ortm (x,y,u), Ortm (x,y,u1)
by ANALOAF:12;
w,w1,v,v1 are_COrtm_wrt x,y
by A1, A3, Th19;
then
Ortm (x,y,w), Ortm (x,y,w1) // v,v1
;
then A6:
v,v1 // Ortm (x,y,w), Ortm (x,y,w1)
by ANALOAF:12;
w,w1,u2,v2 are_COrtm_wrt x,y
by A1, A4, Th19;
then A7:
Ortm (x,y,w), Ortm (x,y,w1) // u2,v2
;
now ( w <> w1 & v <> v1 & not u,u1,u2,v2 are_COrtm_wrt x,y & not v = v1 implies w = w1 )assume that A8:
w <> w1
and A9:
v <> v1
;
( u,u1,u2,v2 are_COrtm_wrt x,y or v = v1 or w = w1 )
Ortm (
x,
y,
w),
Ortm (
x,
y,
w1)
// Ortm (
x,
y,
u),
Ortm (
x,
y,
u1)
by A5, A6, A9, ANALOAF:11;
then
Ortm (
x,
y,
u),
Ortm (
x,
y,
u1)
// u2,
v2
by A1, A7, A8, Th6, ANALOAF:11;
hence
(
u,
u1,
u2,
v2 are_COrtm_wrt x,
y or
v = v1 or
w = w1 )
;
verum end;
hence
( u,u1,u2,v2 are_COrtm_wrt x,y or v = v1 or w = w1 )
; verum