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 a,b,c,d,a1,a2,b1,c1,d1,d2,p,q for Element of DTrSpace(V,w,y);
reserve OTS for OrdTrapSpace;
reserve a,b,c,d for Element of OTS;
reserve a9,b9,c9,d9,a19,b19,d19 for Element of Lambda(OTS);

theorem Th47:
  for x being set holds (x is Element of OTS iff x is Element of Lambda(OTS))
proof
  let x be set;
  Lambda(OTS) = AffinStruct (# the carrier of OTS, lambda(the CONGR of OTS
    ) #) by DIRAF:def 2;
  hence thesis;
end;
