let V be non trivial RealLinearSpace; :: thesis: ( ex y, u, v, w being Element of V st

( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds

( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) implies ex CS being CollProjectiveSpace st

( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) )

given y, u, v, w being Element of V such that A1: ( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds

( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) ; :: thesis: ex CS being CollProjectiveSpace st

( CS = ProjectiveSpace V & CS is at_most-3-dimensional )

( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank & ex p, q1, q2 being Element of (ProjectiveSpace V) st

( not p,q1,q2 are_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st

( r1,r2,r3 are_collinear & q1,q2,q3 are_collinear & p,r3,q3 are_collinear ) ) ) ) by A1, Lm43, Th30;

hence ex CS being CollProjectiveSpace st

( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) by Th31; :: thesis: verum

( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds

( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) implies ex CS being CollProjectiveSpace st

( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) )

given y, u, v, w being Element of V such that A1: ( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds

( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) ; :: thesis: ex CS being CollProjectiveSpace st

( CS = ProjectiveSpace V & CS is at_most-3-dimensional )

( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank & ex p, q1, q2 being Element of (ProjectiveSpace V) st

( not p,q1,q2 are_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st

( r1,r2,r3 are_collinear & q1,q2,q3 are_collinear & p,r3,q3 are_collinear ) ) ) ) by A1, Lm43, Th30;

hence ex CS being CollProjectiveSpace st

( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) by Th31; :: thesis: verum