reserve x,y for object,
  f for Function;

theorem Th42:
  for X,Y being non empty TopSpace for W being open Subset of [:X,
  Y:] for x being Point of X holds Im(W,x) is open Subset of Y
proof
  let X,Y be non empty TopSpace, W be open Subset of [:X,Y:];
  let x be Point of X;
  reconsider W as Relation of the carrier of X, the carrier of Y by
BORSUK_1:def 2;
  reconsider A = W.:{x} as Subset of Y;
  now
    let y be set;
    hereby
      assume y in A;
      then consider z being object such that
A1:   [z,y] in W and
A2:   z in {x} by RELAT_1:def 13;
      consider AA being Subset-Family of [:X,Y:] such that
A3:   W = union AA and
A4:   for e being set st e in AA ex X1 being Subset of X, Y1 being
      Subset of Y st e = [:X1,Y1:] & X1 is open & Y1 is open by BORSUK_1:5;
      z = x by A2,TARSKI:def 1;
      then consider e being set such that
A5:   [x,y] in e and
A6:   e in AA by A1,A3,TARSKI:def 4;
      consider X1 being Subset of X, Y1 being Subset of Y such that
A7:   e = [:X1,Y1:] and
      X1 is open and
A8:   Y1 is open by A4,A6;
      take Y1;
      thus Y1 is open by A8;
A9:   x in X1 by A5,A7,ZFMISC_1:87;
      thus Y1 c= A
      proof
        let z be object;
        assume z in Y1;
        then [x,z] in e by A7,A9,ZFMISC_1:87;
        then
A10:    [x,z] in W by A3,A6,TARSKI:def 4;
        x in {x} by TARSKI:def 1;
        hence thesis by A10,RELAT_1:def 13;
      end;
      thus y in Y1 by A5,A7,ZFMISC_1:87;
    end;
    thus (ex Q being Subset of Y st Q is open & Q c= A & y in Q) implies y in
    A;
  end;
  hence thesis by TOPS_1:25;
end;
