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

theorem Th18:
  for K0a being set,D being non empty Subset of TOP-REAL 2 st K0a=
{p where p is Point of TOP-REAL 2: (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p
  `2) & p<>0.TOP-REAL 2} & D`={0.TOP-REAL 2} holds K0a is non empty Subset of (
  TOP-REAL 2)|D & K0a is non empty Subset of TOP-REAL 2
proof
A1: (1.REAL 2)<>0.TOP-REAL 2 by Lm1,REVROT_1:19;
  let K0a be set,D be non empty Subset of TOP-REAL 2;
  assume that
A2: K0a={p where p is Point of TOP-REAL 2: (p`1<=p`2 & -p`2<=p`1 or p`1
  >=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2} and
A3: D`={0.TOP-REAL 2};
  (1.REAL 2)`1<=(1.REAL 2)`2 & -(1.REAL 2)`2<=(1.REAL 2)`1 or (1.REAL 2)
  `1>=(1.REAL 2)`2 & (1.REAL 2)`1<=-(1.REAL 2)`2 by Th5;
  then
A4: 1.REAL 2 in K0a by A2,A1;
A5: K0a c= D
  proof
    let x be object;
A6: D=D`` .=NonZero TOP-REAL 2 by A3,SUBSET_1:def 4;
    assume x in K0a;
    then
A7: ex p8 being Point of TOP-REAL 2 st x=p8 &( p8`1<=p8`2 & - p8`2<=p8`1 or
    p8`1>=p8`2 & p8`1<=-p8`2)& p8<>0.TOP-REAL 2 by A2;
    then not x in {0.TOP-REAL 2} by TARSKI:def 1;
    hence thesis by A7,A6,XBOOLE_0:def 5;
  end;
  the carrier of (TOP-REAL 2)|D=[#]((TOP-REAL 2)|D) .=D by PRE_TOPC:def 5;
  hence K0a is non empty Subset of ((TOP-REAL 2)|D) by A4,A5;
  thus thesis by A4,A5,XBOOLE_1:1;
end;
