reserve MS for OrtAfPl;
reserve MP for OrtAfSp;
reserve V for RealLinearSpace;
reserve w,y,u,v for VECTOR of V;

theorem
  Gen w,y & MS=AMSpace(V,w,y) implies MS is Euclidean Pappian
  Desarguesian Homogeneous translation Fanoian Moufangian OrtAfPl
proof
  assume that
A1: Gen w,y and
A2: MS=AMSpace(V,w,y);
  reconsider MS9=AMSpace(V,w,y) as OrtAfPl by A2;
A3: MS9 is satisfying_3H by A1,Th20,CONAFFM:6;
  for a,b being Real st a*w + b*y = 0.V
    holds a=0 & b=0 by A1,ANALMETR:def 1;
  then reconsider OAS=OASpace(V) as OAffinSpace by ANALOAF:26;
A4: the AffinStruct of MS9=Lambda(OAS) by ANALMETR:20;
  then
A5: the AffinStruct of MS9 is Moufangian &
    the AffinStruct of MS9 is translational by PAPDESAF:16,17;
  the AffinStruct of MS9 is Pappian & the AffinStruct of
    MS9 is Desarguesian by A4,PAPDESAF:13,14;
  hence thesis by A1,A2,A3,A4,A5,Def2,Def3,Def4,Def5,Def6,Th8,Th22;
end;
