 reserve n,m for Nat,
         p,q for Point of TOP-REAL n, r for Real;
reserve M,M1,M2 for non empty TopSpace;

theorem Th9:
  for N,M be locally_euclidean non empty TopSpace holds
    Fr [:N,M:] = [:[#]N,Fr M:] \/ [:Fr N,[#]M:]
proof
  let N,M be locally_euclidean non empty TopSpace;
  thus Fr [:N,M:] = ([#] [:N,M:]) \Int [:N,M:] by SUBSET_1:def 4
                 .= ([:[#]N,[#]M:]) \Int [:N,M:] by BORSUK_1:def 2
                 .= ([:[#]N,[#]M:])\ [:Int N,Int M:] by Th8
                 .= [:([#]N)\Int N,[#]M:]\/ ([:[#]N,([#]M)\Int M:])
                   by ZFMISC_1:103
                 .= [:([#]N)\Int N,[#]M:]\/ ([:[#]N,Fr M:]) by SUBSET_1:def 4
                 .= [:[#]N,Fr M:]\/[:Fr N,[#]M:] by SUBSET_1:def 4;
end;
