
theorem Th16:
  for X,Z being non empty TopSpace for Y being non empty SubSpace
  of Z holds oContMaps(X, Y) is full SubRelStr of oContMaps(X, Z)
proof
  let X,Z be non empty TopSpace, Y be non empty SubSpace of Z;
  set XY = oContMaps(X, Y), XZ = oContMaps(X, Z);
A1: Omega Y is full SubRelStr of Omega Z by WAYBEL25:17;
A2: [#]Y c= [#]Z by PRE_TOPC:def 4;
  XY is SubRelStr of XZ
  proof
    thus
A3: the carrier of XY c= the carrier of XZ
    proof
      let x be object;
      assume x in the carrier of XY;
      then reconsider f = x as continuous Function of X, Y by Th2;
      dom f = the carrier of X & rng f c= the carrier of Z by A2,FUNCT_2:def 1;
      then x is continuous Function of X,Z by FUNCT_2:2,PRE_TOPC:26;
      then x is Element of XZ by Th2;
      hence thesis;
    end;
    let x,y be object;
    assume
A4: [x,y] in the InternalRel of XY;
    reconsider x, y as Element of XY by A4,ZFMISC_1:87;
    reconsider a = x, b = y as Element of XZ by A3;
    reconsider f = x, g = y as continuous Function of X, Omega Y by Th1;
    reconsider f9 = a, g9 = b as continuous Function of X, Omega Z by Th1;
    x <= y by A4,ORDERS_2:def 5;
    then f <= g by Th3;
    then f9 <= g9 by A1,YELLOW16:2;
    then a <= b by Th3;
    hence thesis by ORDERS_2:def 5;
  end;
  then reconsider XY as non empty SubRelStr of XZ;
A5: (the InternalRel of XZ)|_2 the carrier of XY = (the InternalRel of XZ)
  /\ [:the carrier of XY, the carrier of XY:] by WELLORD1:def 6;
  the InternalRel of XY = ((the InternalRel of XZ)|_2 the carrier of XY )
  qua set
  proof
    the InternalRel of XY c= the InternalRel of XZ by YELLOW_0:def 13;
    hence
    the InternalRel of XY c= (the InternalRel of XZ)|_2 the carrier of XY
    by A5,XBOOLE_1:19;
    let x,y be object;
    assume
A6: [x,y] in (the InternalRel of XZ)|_2 the carrier of XY;
    then
A7: [x,y] in [:the carrier of XY, the carrier of XY:] by A5,XBOOLE_0:def 4;
    reconsider x,y as Element of XY by A7,ZFMISC_1:87;
    the carrier of XY c= the carrier of XZ by YELLOW_0:def 13;
    then reconsider a = x, b = y as Element of XZ;
    reconsider f9 = a, g9 = b as continuous Function of X, Omega Z by Th1;
    reconsider f = x, g = y as continuous Function of X, Omega Y by Th1;
    [a,b] in the InternalRel of XZ by A5,A6,XBOOLE_0:def 4;
    then a <= b by ORDERS_2:def 5;
    then f9 <= g9 by Th3;
    then f <= g by A1,YELLOW16:3;
    then x <= y by Th3;
    hence thesis by ORDERS_2:def 5;
  end;
  hence thesis by YELLOW_0:def 14;
end;
