reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;

theorem Th97:
  for X0 being non empty SubSpace of X st the carrier of X0
  misses A for x0 being Point of X0 holds (modid(X,A))|X0 is_continuous_at x0
proof
  let X0 be non empty SubSpace of X;
  assume
A1: (the carrier of X0) /\ A = {};
  let x0 be Point of X0;
  x0 in the carrier of X0 & the carrier of X0 c= the carrier of X by BORSUK_1:1
;
  then reconsider x = x0 as Point of X;
  not x in A by A1,XBOOLE_0:def 4;
  hence thesis by Th58,Th96;
end;
