reserve p,q for Point of TOP-REAL 2;

theorem Th28:
  for P being compact non empty Subset of TOP-REAL 2 st P={q: |.q
  .|=1} holds W-bound(P)=-1 & E-bound(P)=1 & S-bound(P)=-1 & N-bound(P)=1
proof
  let P be compact non empty Subset of TOP-REAL 2;
A1: the carrier of ((TOP-REAL 2)|P) =P by PRE_TOPC:8;
  assume
A2: P={q: |.q.|=1};
  hence W-bound(P)=-1 by Lm4;
  proj1.:P=[.-1,1.] by A2,Lm3;
  then (proj1|P).:P=[.-1,1.] by RELAT_1:129;
  then upper_bound ((proj1|P).:the carrier of ((TOP-REAL 2)|P))=1
  by A1,JORDAN5A:19;
  then upper_bound (proj1|P)=1 by PSCOMP_1:def 2;
  hence E-bound P=1 by PSCOMP_1:def 9;
  proj2.:P=[.-1,1.] by A2,Lm3;
  then
A3: (proj2|P).:P=[.-1,1.] by RELAT_1:129;
  then lower_bound ((proj2|P).:P)=-1 by JORDAN5A:19;
  then lower_bound (proj2|P)=-1 by A1,PSCOMP_1:def 1;
  hence S-bound P=-1 by PSCOMP_1:def 10;
  upper_bound ((proj2|P).:P)=1 by A3,JORDAN5A:19;
  then upper_bound (proj2|P)=1 by A1,PSCOMP_1:def 2;
  hence thesis by PSCOMP_1:def 8;
end;
