reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;
reserve D for non vertical non horizontal non empty compact Subset of TOP-REAL
  2;

theorem Th80:
  UBD (L~SpStSeq D)<>{} & UBD (L~SpStSeq D)
is_outside_component_of (L~SpStSeq D) & BDD (L~SpStSeq D)<>{} & BDD (L~SpStSeq
  D) is_inside_component_of (L~SpStSeq D)
proof
  set f=SpStSeq D;
A1: UBD (L~SpStSeq D)=LeftComp (SpStSeq D) by Th79;
  hence UBD (L~SpStSeq D)<>{};
  LeftComp f is_a_component_of (L~f)` & not LeftComp f is bounded by Th74,
GOBOARD9:def 1;
  hence UBD (L~SpStSeq D) is_outside_component_of (L~SpStSeq D) by A1;
A2: BDD (L~SpStSeq D)=RightComp (SpStSeq D) by Th79;
  hence BDD (L~SpStSeq D)<>{};
  RightComp (SpStSeq D) is_a_component_of (L~f)` & RightComp (SpStSeq D)
  is bounded by Th78,GOBOARD9:def 2;
  hence thesis by A2;
end;
