reserve V for RealLinearSpace;
reserve x,y for VECTOR of V;

theorem Th4:
  Gen x,y implies CESpace(V,x,y) is Oriented_Orthogonality_Space
proof
  assume
A1: Gen x,y;
  then
A2: for u,v,w being Element of CESpace(V,x,y) holds ex u1 being Element of
  CESpace(V,x,y) st w<>u1 & w,u1 '//' u,v by Th1;
A3: for u,v,w being Element of CESpace(V,x,y) holds ex u1 being Element of
  CESpace(V,x,y) st w<>u1 & u,v '//' w,u1 by A1,Th1;
  for u,u1,v,v1,w,w1,w2 being Element of CESpace(V,x,y) holds u,u '//' v,w
& u,v '//' w,w & (u,v '//' u1,v1 & u,v '//' v1,u1 implies u=v or u1=v1) & (u,v
'//' u1,v1 & u,v '//' u1,w implies u,v '//' v1,w or u,v '//' w,v1) & (u, v '//'
  u1,v1 implies v,u '//' v1,u1) & (u,v '//' u1,v1 & u,v '//' v1,w implies u,v
  '//' u1,w) & (u,u1 '//' v,v1 implies v,v1 '//' u,u1 or v,v1 '//' u1,u) by A1
,Th1;
  hence thesis by A2,A3,Def1;
end;
