
theorem Th28:
  for T being topological_semilattice non empty TopSpace-like
  TopRelStr, x being Element of T holds x"/\" is continuous
proof
  let T be topological_semilattice non empty TopSpace-like TopRelStr, x be
  Element of T;
  let p be Point of T, G be a_neighborhood of x"/\".p;
  set TT = [:T,T qua TopSpace:];
A1: the carrier of TT = [:the carrier of T,the carrier of T:]
       by BORSUK_1:def 2;
  then reconsider xp = [x,p] as Point of TT by YELLOW_3:def 2;
  the carrier of [:T,T:] = [:the carrier of T,the carrier of T:] by
YELLOW_3:def 2;
  then reconsider f = inf_op T as Function of TT, T by A1;
A2: f.xp = f.(x,p) .= x "/\" p by WAYBEL_2:def 4;
  G is a_neighborhood of x"/\"p & f is continuous by Def5,WAYBEL_1:def 18;
  then consider H being a_neighborhood of xp such that
A3: f.:H c= G by A2;
  consider A being Subset-Family of TT such that
A4: Int H = union A and
A5: 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;
  xp in Int H by CONNSP_2:def 1;
  then consider Z being set such that
A6: xp in Z and
A7: Z in A by A4,TARSKI:def 4;
  consider X1, Y1 being Subset of T such that
A8: Z = [:X1,Y1:] and
  X1 is open and
A9: Y1 is open by A5,A7;
  p in Y1 by A6,A8,ZFMISC_1:87;
  then reconsider Y1 as a_neighborhood of p by A9,CONNSP_2:3;
  take Y1;
  let b be object;
  assume b in (x"/\").:Y1;
  then consider a being object such that
  a in dom (x"/\") and
A10: a in Y1 and
A11: b = x"/\".a by FUNCT_1:def 6;
  reconsider a as Element of T by A10;
  x in X1 by A6,A8,ZFMISC_1:87;
  then [x,a] in Z by A8,A10,ZFMISC_1:87;
  then
A12: [x,a] in Int H by A4,A7,TARSKI:def 4;
  [x,a] in [:the carrier of T,the carrier of T:] by ZFMISC_1:87;
  then
A13: Int H c= H & [x,a] in dom f by A1,FUNCT_2:def 1,TOPS_1:16;
  b = x "/\" a by A11,WAYBEL_1:def 18
    .= f.(x,a) by WAYBEL_2:def 4;
  then b in f.:H by A12,A13,FUNCT_1:def 6;
  hence thesis by A3;
end;
