let G be IncProjStr ; ( G is configuration iff for p, q being POINT of G
for P, Q being LINE of G st p on P,Q & q on P,Q & not p = q holds
P = Q )
thus
( G is configuration implies for p, q being POINT of G
for P, Q being LINE of G st p on P,Q & q on P,Q & not p = q holds
P = Q )
; ( ( for p, q being POINT of G
for P, Q being LINE of G st p on P,Q & q on P,Q & not p = q holds
P = Q ) implies G is configuration )
hereby verum
assume A1:
for
p,
q being
POINT of
G for
P,
Q being
LINE of
G st
p on P,
Q &
q on P,
Q & not
p = q holds
P = Q
;
G is configuration now for p, q being POINT of G
for P, Q being LINE of G st p on P & q on P & p on Q & q on Q & not p = q holds
P = Qlet p,
q be
POINT of
G;
for P, Q being LINE of G st p on P & q on P & p on Q & q on Q & not p = q holds
P = Qlet P,
Q be
LINE of
G;
( p on P & q on P & p on Q & q on Q & not p = q implies P = Q )assume
(
p on P &
q on P &
p on Q &
q on Q )
;
( p = q or P = Q )then
(
p on P,
Q &
q on P,
Q )
;
hence
(
p = q or
P = Q )
by A1;
verum end; hence
G is
configuration
;
verum
end;