reserve A, B, X, Y for set;

theorem
  for L1, L2 being RelStr, A being Subset of L1, J being Subset of L2 st
  the RelStr of L1 = the RelStr of L2 & A = J holds subrelstr A = subrelstr J
proof
  let L1, L2 be RelStr, A be Subset of L1, J be Subset of L2 such that
A1: the RelStr of L1 = the RelStr of L2 and
A2: A = J;
A3: the carrier of subrelstr A = A by YELLOW_0:def 15
    .= the carrier of subrelstr J by A2,YELLOW_0:def 15;
  then
  the InternalRel of subrelstr A = (the InternalRel of L2)|_2(the carrier
  of subrelstr J) by A1,YELLOW_0:def 14
    .= the InternalRel of subrelstr J by YELLOW_0:def 14;
  hence thesis by A3;
end;
