consider V being RealLinearSpace such that
A1: ex w, y being VECTOR of V st Gen w,y by ANALMETR:3;
consider w, y being VECTOR of V such that
A2: Gen w,y by A1;
reconsider MS = AMSpace (V,w,y) as OrtAfPl by A2, ANALMETR:34;
A3: MS is satisfying_3H by A2, Th20, CONAFFM:6;
for a, b being Real st (a * w) + (b * y) = 0. V holds
( a = 0 & b = 0 ) by A2, ANALMETR:def 1;
then reconsider OAS = OASpace V as OAffinSpace by ANALOAF:26;
take MS ; :: thesis: ( MS is Euclidean & MS is Pappian & MS is Desarguesian & MS is Homogeneous & MS is translation & MS is Fanoian & MS is Moufangian )
A4: AffinStruct(# the carrier of MS, the CONGR of MS #) = Lambda OAS by ANALMETR:20;
then A5: ( AffinStruct(# the carrier of MS, the CONGR of MS #) is Moufangian & AffinStruct(# the carrier of MS, the CONGR of MS #) is translational ) by PAPDESAF:16, PAPDESAF:17;
( AffinStruct(# the carrier of MS, the CONGR of MS #) is Pappian & AffinStruct(# the carrier of MS, the CONGR of MS #) is Desarguesian ) by A4, PAPDESAF:13, PAPDESAF:14;
hence ( MS is Euclidean & MS is Pappian & MS is Desarguesian & MS is Homogeneous & MS is translation & MS is Fanoian & MS is Moufangian ) by A2, A3, A4, A5, Th8, Th22; :: thesis: verum