consider V being non trivial RealLinearSpace such that
A1: ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) by Th16;
consider u, v, w being Element of V such that
A2: for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) and
for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) by A1;
reconsider V = V as up-3-dimensional RealLinearSpace by A2, Def6;
take CS = ProjectiveSpace V; :: thesis: ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian & CS is 2-dimensional )
thus ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian ) ; :: thesis: CS is 2-dimensional
consider CS1 being CollProjectiveSpace such that
A3: CS1 = ProjectiveSpace V and
A4: CS1 is 2-dimensional by A1, Th31;
thus CS is 2-dimensional by A3, A4; :: thesis: verum