consider V being non trivial RealLinearSpace such that
A5: 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 Th23;
reconsider V = V as up-3-dimensional RealLinearSpace by A5, Lm43;
take CS = ProjectiveSpace V; :: thesis: ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian & CS is at_most-3-dimensional & not CS is 2-dimensional )
thus ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian ) ; :: thesis: ( CS is at_most-3-dimensional & not CS is 2-dimensional )
ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & not CS is 2-dimensional & CS is at_most-3-dimensional ) by A5, Th36;
hence ( CS is at_most-3-dimensional & not CS is 2-dimensional ) ; :: thesis: verum