
theorem Th9:
  for S being non empty reflexive RelStr, D being non empty Subset of S holds
  Net-Str D = Net-Str (D, (id the carrier of S)|D)
proof
  let S be non empty reflexive RelStr;
  let D be non empty Subset of S;
  set M = Net-Str (D, (id the carrier of S)|D);
A1: D = the carrier of M by WAYBEL11:def 10;
A2: (id the carrier of S)|D = the mapping of M by WAYBEL11:def 10;
A3: (id the carrier of S)|D = id D by FUNCT_3:1;
A4: (the InternalRel of S)|_2 D c= the InternalRel of S by XBOOLE_1:17;
  now
    let x, y be object;
    hereby
      assume
A5:   [x,y] in (the InternalRel of S)|_2 D;
      then
A6:   x in D by ZFMISC_1:87;
A7:   y in D by A5,ZFMISC_1:87;
      reconsider x9 = x, y9 = y as Element of M by A1,A5,ZFMISC_1:87;
A8:   x9 = ((id the carrier of S)|D).x9 by A3,A6,FUNCT_1:18;
      y9 = ((id the carrier of S)|D).y9 by A3,A7,FUNCT_1:18;
      then M.x9 <= M.y9 by A2,A4,A5,A8,ORDERS_2:def 5;
      then x9 <= y9 by WAYBEL11:def 10;
      hence [x,y] in the InternalRel of M by ORDERS_2:def 5;
    end;
    assume
A9: [x,y] in the InternalRel of M;
    then reconsider x9 = x, y9 = y as Element of M by ZFMISC_1:87;
    x9 <= y9 by A9,ORDERS_2:def 5;
    then M.x9 <= M.y9 by WAYBEL11:def 10;
    then
A10: [M.x9, M.y9] in the InternalRel of S by ORDERS_2:def 5;
A11: x9 = ((id the carrier of S)|D).x9 by A1,A3;
    y9 = ((id the carrier of S)|D). y9 by A1,A3;
    hence [x,y] in (the InternalRel of S)|_2 D by A1,A2,A9,A10,A11,
XBOOLE_0:def 4;
  end;
  hence thesis by A1,A2,RELAT_1:def 2;
end;
