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 Th91:
  for G being non empty TopSpace,A,B,C,D being Subset of G st B
  is a_component & C is a_component & A \/ B=the carrier of G & C
  misses A holds C=B
proof
  let G be non empty TopSpace,A,B,C,D be Subset of G;
  assume that
A1: B is a_component and
A2: C is a_component and
A3: A \/ B=the carrier of G and
A4: C misses A;
  now
    C /\ (the carrier of G)=C by XBOOLE_1:28;
    then
A5: C /\ A \/ C /\ B=C by A3,XBOOLE_1:23;
    assume C misses B;
    then
A6: C /\ B = {};
    C <> {}G by A2,CONNSP_1:32;
    hence contradiction by A4,A6,A5;
  end;
  hence thesis by A1,A2,CONNSP_1:35;
end;
