reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;

theorem Th38:
  delta the carrier of T is continuous Function of T, [:T,T:]
proof
  the carrier of [:T,T:] = [: the carrier of T, the carrier of T:] by
BORSUK_1:def 2;
  then reconsider f = delta the carrier of T as Function of T, [:T,T:];
  f is continuous
  proof
    let W be Point of T, G be a_neighborhood of f.W;
    consider A being Subset-Family of [:T,T:] such that
A1: Int G = union A and
A2: for e being set st e in A ex X1, Y1 being Subset of T st e = [:X1,
    Y1:] & X1 is open & Y1 is open by BORSUK_1:5;
    f.W in Int G by CONNSP_2:def 1;
    then consider Z being set such that
A3: f.W in Z and
A4: Z in A by A1,TARSKI:def 4;
    consider X1, Y1 being Subset of T such that
A5: Z = [:X1,Y1:] and
A6: X1 is open & Y1 is open by A2,A4;
    f.W = [W,W] by FUNCT_3:def 6;
    then W in X1 & W in Y1 by A3,A5,ZFMISC_1:87;
    then
A7: W in X1 /\ Y1 by XBOOLE_0:def 4;
    X1 /\ Y1 is open by A6;
    then W in Int (X1 /\ Y1) by A7,TOPS_1:23;
    then reconsider H = X1 /\ Y1 as a_neighborhood of W by CONNSP_2:def 1;
A8: f.:H c= Int G
    proof
      let y be object;
      assume y in f.:H;
      then consider x being object such that
A9:   x in dom f and
A10:  x in H and
A11:  y = f.x by FUNCT_1:def 6;
A12:  x in X1 & x in Y1 by A10,XBOOLE_0:def 4;
      y = [x,x] by A9,A11,FUNCT_3:def 6;
      then y in Z by A5,A12,ZFMISC_1:87;
      hence thesis by A1,A4,TARSKI:def 4;
    end;
    take H;
    Int G c= G by TOPS_1:16;
    hence thesis by A8;
  end;
  hence thesis;
end;
