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 OrtAfSp
proof
  set POS = AMSpace(V,w,y);
  set X = AffinStruct(#the carrier of POS,the CONGR of POS#);
  assume
A1: Gen w,y;
  then
A2: for a,b,c be Element of POS holds ex x being Element of POS st a,b _|_ c
  ,x & c <>x by Th27;
A3: X = Lambda(OASpace(V)) by Th20;
  for a,b being Real st a*w + b*y = 0.V holds a=0 & b=0 by A1;
  then OASpace(V) is OAffinSpace by ANALOAF:26;
  then
A4: X is AffinSpace by A3,DIRAF:41;
  ( 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 _|_ p,s implies a
,b _|_ q,s))& for a,b,c be Element of POS holds a<>b implies ex x being Element
  of POS st a,b // a,x & a,b _|_ x,c by A1,Th23,Th24,Th25,Th26,Th29,Th30,Th32;
  hence thesis by A2,A4,Def7;
end;
