
theorem Th17:
  for T being non empty TopSpace, S being non empty SubSpace of T
  holds Omega S is full SubRelStr of Omega T
proof
  let T be non empty TopSpace, S be non empty SubSpace of T;
A1: the carrier of S c= the carrier of T by BORSUK_1:1;
A2: the carrier of Omega S = the carrier of S by Lm1;
A3: the InternalRel of Omega S c= the InternalRel of Omega T
  proof
    let x, y be object;
    assume
A4: [x,y] in the InternalRel of Omega S;
    then reconsider o1 = x, o2 = y as Element of Omega S by ZFMISC_1:87;
A5: y in the carrier of Omega S by A4,ZFMISC_1:87;
    then reconsider s2 = y as Element of S by Lm1;
    x in the carrier of Omega S by A4,ZFMISC_1:87;
    then reconsider o3 = x, o4 = y as Element of Omega T by A1,A2,A5,Lm1;
    reconsider t2 = y as Element of T by A1,A2,A5;
    Cl {s2} = (Cl {t2}) /\ ([#]S) by PRE_TOPC:17;
    then
A6: Cl {s2} c= Cl {t2} by XBOOLE_1:17;
    o1 <= o2 by A4;
    then ex Y2 being Subset of S st Y2 = {o2} & o1 in Cl Y2 by Def2;
    then o3 <= o4 by A6,Def2;
    hence thesis;
  end;
A7: the InternalRel of Omega S = (the InternalRel of Omega T)|_2 the
  carrier of Omega S
  proof
    let x, y be object;
    thus [x,y] in the InternalRel of Omega S implies [x,y] in (the InternalRel
    of Omega T)|_2 the carrier of Omega S by A3,XBOOLE_0:def 4;
    assume
A8: [x,y] in (the InternalRel of Omega T)|_2 the carrier of Omega S;
    then
A9: y in the carrier of Omega S by ZFMISC_1:87;
A10: x in the carrier of Omega S by A8,ZFMISC_1:87;
    then reconsider t1 = x, t2 = y as Element of T by A1,A2,A9;
    reconsider o1 = x, o2 = y as Element of Omega T by A1,A2,A10,A9,Lm1;
    [x,y] in the InternalRel of Omega T by A8,XBOOLE_0:def 4;
    then o1 <= o2;
    then
A11: ex Y being Subset of T st Y = {t2} & t1 in Cl Y by Def2;
    reconsider s1 = x, s2 = y as Element of S by A10,A9,Lm1;
    reconsider o3 = x, o4 = y as Element of Omega S by A8,ZFMISC_1:87;
    Cl {s2} = (Cl {t2}) /\ [#]S by PRE_TOPC:17;
    then s1 in Cl {s2} by A11,XBOOLE_0:def 4;
    then o3 <= o4 by Def2;
    hence thesis;
  end;
  the carrier of Omega S c= the carrier of Omega T by A1,A2,Lm1;
  hence thesis by A3,A7,YELLOW_0:def 13,def 14;
end;
