
theorem Th32:
  for S being non empty reflexive RelStr, D being non empty Subset of S
  holds the mapping of Net-Str D = id D & the carrier of Net-Str D = D &
  Net-Str D is full SubRelStr of S
proof
  let S be non empty reflexive RelStr, D be non empty Subset of S;
  set N = Net-Str D;
A1: dom id D = D;
  rng id D = D;
  then reconsider g = id D as Function of D, the carrier of S
  by A1,FUNCT_2:def 1,RELSET_1:4;
  (id the carrier of S)|D = id D by FUNCT_3:1;
  then
A2: N = NetStr (#D, (the InternalRel of S)|_2 D, g#) by WAYBEL17:def 4;
  then the InternalRel of N c= the InternalRel of S by XBOOLE_1:17;
  hence thesis by A2,YELLOW_0:def 13,def 14;
end;
