consider V being non trivial RealLinearSpace such that
A2:
ex u, v, w, u1 being Element of V st
( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
by Th22;
reconsider V = V as up-3-dimensional RealLinearSpace by A2, Lm42;
take CS = ProjectiveSpace V; ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian & CS is at_most-3-dimensional & CS is up-3-dimensional )
thus
( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian )
; ( CS is at_most-3-dimensional & CS is up-3-dimensional )
ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is up-3-dimensional & CS is at_most-3-dimensional )
by A2, Th34;
hence
( CS is at_most-3-dimensional & CS is up-3-dimensional )
; verum