reserve V for RealLinearSpace;
reserve x,y for VECTOR of V;
reserve AS for Oriented_Orthogonality_Space;
reserve u,u1,u2,u3,v,v1,v2,v3,w,w1 for Element of AS;

theorem
  u,u1 // v,v1 implies u1,u // v1,v
proof
  assume u,u1 // v,v1;
  then consider w,w1 such that
A1: w<>w1 and
A2: w,w1 '//' u,u1 and
A3: w,w1 '//' v,v1;
A4: w1,w '//' v1,v by A3,Def1;
  w1,w '//' u1,u by A2,Def1;
  hence thesis by A1,A4;
end;
