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

theorem Th39:
  pr1(the carrier of S,the carrier of T) is continuous Function of [:S,T:], S
proof
  set I = the carrier of S, J = the carrier of T;
A1: the carrier of [:S,T:] = [:I,J:] by BORSUK_1:def 2;
  then reconsider f = pr1(I,J) as Function of [:S,T:], S;
  f is continuous
  proof
    let w be Point of [:S,T:], G be a_neighborhood of f.w;
    set H = [:Int G, [#]T:];
A2: Int H = [:Int Int G, Int [#]T:] by BORSUK_1:7
      .= [:Int G, [#]T:] by TOPS_1:15;
    consider x, y being object such that
A3: x in I and
A4: y in J and
A5: w = [x,y] by A1,ZFMISC_1:def 2;
    f.w in Int G & f.(x,y) = x by A3,A4,CONNSP_2:def 1,FUNCT_3:def 4;
    then w in Int H by A4,A5,A2,ZFMISC_1:def 2;
    then reconsider H as a_neighborhood of w by CONNSP_2:def 1;
    take H;
    reconsider X = Int G as non empty Subset of S by CONNSP_2:def 1;
    [:X,[#]T:] <> {};
    then f.:H = Int G by EQREL_1:49;
    hence thesis by TOPS_1:16;
  end;
  hence thesis;
end;
