reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;

theorem Th34:
  Int A = Int NTop2Top A
  proof
    set NA = A;
    reconsider T = NTop2Top NT as non empty TopSpace;
    reconsider A9 = NTop2Top NA as Subset of T;
    now
      now
        let o be object;
        assume o in Int NA;
        then consider y be Point of NT such that
A1:     o = y and
A2:     y is_interior_point_of NA;
        consider NO be open Subset of NT such that
A3:     y in NO and
A4:     NO c= NA by A2,Lm4;
        reconsider O = NO as open Subset of T by Lm9;
        y in O by A3;
        then reconsider y9 = y as Point of T;
        y9 in O & O c= A9 by A3,A4;
        then A is a_neighborhood of y9 by URYSOHN1:def 6;
        hence o in Int A9 by CONNSP_2:def 1,A1;
      end;
      hence Int NA c= Int A9;
      now
        let o be object;
        assume
A5:     o in Int A9;
        then reconsider x = o as Point of T;
        consider O be Subset of T such that
A6:     O is open and
A7:     x in O and
A8:     O c= A9 by A5,CONNSP_2:def 1,URYSOHN1:def 6;
        Top2NTop T = NT by FINTOPO7:25;
        then reconsider NO = O as open Subset of NT
          by A6,Lm1;
        x in NO by A7;
        then reconsider y = x as Point of NT;
        y in NO & NO c= NA by A7,A8;
        then y is_interior_point_of NA by Lm4;
        hence o in Int NA;
      end;
      hence Int A9 c= Int NA;
    end;
    hence thesis;
  end;
