reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);

theorem
  Gen w,y implies AMSpace(V,w,y) is OrtAfPl
proof
  set POS = AMSpace(V,w,y);
  set X = AffinStruct(#the carrier of POS,the CONGR of POS#);
A1: X = Lambda(OASpace(V)) by Th20;
  assume
A2: Gen w,y;
  then
  ( for a,b being Real st a*w + b*y = 0.V holds a=0 & b=0)&
  for w1 ex a,b being Real st w1 = a*w+b*y;
  then OASpace(V) is OAffinPlane by ANALOAF:28;
  then
A3: X is AffinPlane by A1,DIRAF:45;
  ( for a,b,c,d,p,q,r,s be Element of POS holds (a,b _|_ a,b implies a=b)
& a,b _|_ c,c & (a,b _|_ c,d implies a,b _|_ d,c & c,d _|_ a,b) & (a,b _|_ p,q
& a,b // r,s implies p,q _|_ r,s or a=b) & (a,b _|_ p,q & a,b _|_ r,s implies p
,q // r,s or a=b))& for a,b,c be Element of POS holds ex x being Element of POS
  st a,b _|_ c,x & c <>x by A2,Th23,Th24,Th25,Th26,Th27,Th28,Th30;
  hence thesis by A3,Def8;
end;
