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;

theorem Th71:
  X1 is SubSpace of X0 implies (f|X0)|X1 = f|X1
proof
  assume
A1: X1 is SubSpace of X0;
  then
A2: the carrier of X1 c= the carrier of X0 by TSEP_1:4;
  f|X0 = f|(the carrier of X0);
  then reconsider h = f|(the carrier of X0) as Function of X0,Y;
  thus (f|X0)|X1 = h|(the carrier of X1) by A1,Def5
    .= f|X1 by A2,FUNCT_1:51;
end;
