reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th15:
  for D being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL 2
)|D st K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2}
  holds rng (Out_In_Sq|K0) c= the carrier of ((TOP-REAL 2)|D)|K0
proof
  let D be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|D;
A1: the carrier of ((TOP-REAL 2)|D) =[#]((TOP-REAL 2)|D)
    .=D by PRE_TOPC:def 5;
  then reconsider K00=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1;
  assume
A2: K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2};
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K00
  holds q`1<>0
  proof
    let q be Point of TOP-REAL 2;
A4: the carrier of (TOP-REAL 2)|K00=[#]((TOP-REAL 2)|K00)
      .=K0 by PRE_TOPC:def 5;
    assume q in the carrier of (TOP-REAL 2)|K00;
    then
A5: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
    or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A2,A4;
    now
      assume
A6:   q`1=0;
      then q`2=0 by A5;
      hence contradiction by A5,A6,EUCLID:53,54;
    end;
    hence thesis;
  end;
  let y be object;
  assume y in rng (Out_In_Sq|K0);
  then consider x being object such that
A7: x in dom (Out_In_Sq|K0) and
A8: y=(Out_In_Sq|K0).x by FUNCT_1:def 3;
A9: x in (dom Out_In_Sq) /\ K0 by A7,RELAT_1:61;
  then
A10: x in K0 by XBOOLE_0:def 4;
  K0 c= the carrier of TOP-REAL 2 by A1,XBOOLE_1:1;
  then reconsider p=x as Point of TOP-REAL 2 by A10;
A11: Out_In_Sq.p=y by A8,A10,FUNCT_1:49;
A12: ex px being Point of TOP-REAL 2 st x=px &( px`2<=px`1 & - px`1<=px`2 or
  px`2>=px`1 & px`2<=-px`1)& px<>0.TOP-REAL 2 by A2,A10;
  then
A13: Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by Def1;
  set p9=|[1/p`1,p`2/p`1/p`1]|;
  K00=[#]((TOP-REAL 2)|K00) by PRE_TOPC:def 5
    .=the carrier of ((TOP-REAL 2)|K00);
  then
A14: p in the carrier of ((TOP-REAL 2)|K00) by A9,XBOOLE_0:def 4;
A15: p9`1=1/p`1 by EUCLID:52;
A16: now
    assume p9=0.TOP-REAL 2;
    then 0 *p`1=1/p`1*p`1 by A15,EUCLID:52,54;
    hence contradiction by A14,A3,XCMPLX_1:87;
  end;
A17: p`1<>0 by A14,A3;
  now
    per cases;
    suppose
A18:  p`1>=0;
      then p`2/p`1<=p`1/p`1 & (-1 *p`1)/p`1<=p`2/p`1 or p`2>=p`1 & p`2<=-1 *p
      `1 by A12,XREAL_1:72;
      then
A19:  p`2/p`1<=1 & (-1)*p`1/p`1<=p`2/p`1 or p`2>=p`1 & p`2<=-1 *p`1 by A14,A3,
XCMPLX_1:60;
      then p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=1 & p`2/p`1<=(-1)*p`1/p`1 by
A17,A18,XCMPLX_1:89;
      then (-1)/p`1<= p`2/p`1/p`1 by A18,XREAL_1:72;
      then
A20:  p`2/p`1/p`1 <=1/p`1 & -(1/p`1)<= p`2/p`1/p`1 or p`2/p`1/p`1 >=1/p`1
      & p`2/p`1/p`1<= -(1/p`1) by A17,A18,A19,XREAL_1:72;
      p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:52;
      hence y in K0 by A2,A11,A16,A13,A20;
    end;
    suppose
A21:  p`1<0;
A22:   now
      per cases by A12;
      case that
A23:    p`2<=p`1 and
A24: (-1 *p`1)<=p`2;
      p`2/p`1 >=1 by A21,A23,XREAL_1:182;
     hence p`2/p`1/p`1 <=1/p`1 by A21,XREAL_1:73;
      (-1) *p`1<=p`2 by A24;
      then -1 >= p`2/p`1 by A21,XREAL_1:78;
      then (-1)/p`1<= p`2/p`1/p`1 by A21,XREAL_1:73;
     hence -(1/p`1)<= p`2/p`1/p`1;
    end;
      case that
A25:   p`2>=p`1 and
A26:  p`2<=-1 *p`1;
      p`2/p`1 <=1 by A21,A25,XREAL_1:186;
     hence p`2/p`1/p`1 >=1/p`1 by A21,XREAL_1:73;
      (-1) *p`1>=p`2 by A26;
      then -1 <= p`2/p`1 by A21,XREAL_1:80;
      then (-1)/p`1>= p`2/p`1/p`1 by A21,XREAL_1:73;
     hence -(1/p`1)>= p`2/p`1/p`1;
    end;
    end;
      p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:52;
      hence y in K0 by A2,A11,A16,A13,A22;
    end;
  end;
  then y in [#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
  hence thesis;
end;
