
theorem
  for R being non empty RelStr, X being non empty Subset of R holds
  the carrier of X+id = X & X+id is full SubRelStr of R &
  for x being Element of X+id holds X+id.x = x
proof
  let R be non empty RelStr, X be non empty Subset of R;
A1: the RelStr of X+id = the RelStr of (subrelstr X)+id by WAYBEL_9:def 8;
A2: the mapping of X+id = incl(subrelstr X, R)*the mapping of (subrelstr X)
  +id by WAYBEL_9:def 8;
A3: the carrier of subrelstr X = X by YELLOW_0:def 15;
  hence
A4: the carrier of X+id = X by A1,WAYBEL_9:def 5;
A5: the RelStr of (subrelstr X)+id = subrelstr X by WAYBEL_9:def 5;
  the InternalRel of subrelstr X c= the InternalRel of R by YELLOW_0:def 13;
  then reconsider S = X+id as SubRelStr of R by A1,A3,A5,YELLOW_0:def 13;
  the InternalRel of S = (the InternalRel of R)|_2 the carrier of S
  by A1,A5,YELLOW_0:def 14;
  hence X+id is full SubRelStr of R by YELLOW_0:def 14;
  let x be Element of X+id;
  id subrelstr X = id X by YELLOW_0:def 15;
  then
A6: the mapping of (subrelstr X)+id = id X by WAYBEL_9:def 5;
A7: dom id X = X;
  incl(subrelstr X, R) = id X by A3,Def1;
  then the mapping of X+id = id X by A2,A6,A7,RELAT_1:52;
  hence thesis by A4;
end;
