reserve i for Integer,
  a, b, r, s for Real;

theorem
  for S, T being non empty TopSpace, Z being non empty SubSpace of T, f
being Function of S,T, g being Function of S,Z st f = g & f is open holds g is
  open
proof
  let S, T be non empty TopSpace, Z be non empty SubSpace of T, f be Function
  of S,T, g be Function of S,Z such that
A1: f = g and
A2: f is open;
  for p being Point of S, P being open a_neighborhood of p ex R being
  a_neighborhood of g.p st R c= g.:P
  proof
    let p be Point of S, P be open a_neighborhood of p;
    consider R being open a_neighborhood of f.p such that
A3: R c= f.:P by A2,TOPGRP_1:22;
    reconsider R2 = R /\ [#]Z as Subset of Z;
    reconsider R2 as a_neighborhood of g.p by A1,Th5;
    take R2;
    R2 c= R by XBOOLE_1:17;
    hence thesis by A1,A3;
  end;
  hence thesis by TOPGRP_1:23;
end;
