
theorem Th27:
  for X being non empty TopSpace, x being Point of X for Y being
  monotone-convergence T_0-TopSpace ex F being directed-sups-preserving
  projection Function of oContMaps(X,Y), oContMaps(X,Y) st (for f being
  continuous Function of X,Y holds F.f = X --> (f.x)) & ex h being continuous
  Function of X,X st h = X --> x & F = oContMaps(h, Y)
proof
  let X be non empty TopSpace, x be Point of X;
  let Y be monotone-convergence T_0-TopSpace;
  set XY = oContMaps(X,Y);
  reconsider f = X --> x as continuous Function of X,X;
  set F = oContMaps(f, Y);
  dom f = the carrier of X by FUNCT_2:def 1;
  then f*f = (the carrier of X) --> (f.x) by FUNCOP_1:17
    .= f by FUNCOP_1:7;
  then f is idempotent by QUANTAL1:def 9;
  then F is directed-sups-preserving idempotent Function of XY,XY by Th11,Th15;
  then reconsider
  F as directed-sups-preserving projection Function of XY,XY by WAYBEL_1:def 13
;
  take F;
  hereby
    let h be continuous Function of X,Y;
A1: the carrier of X = dom h by FUNCT_2:def 1;
    thus F.h = h*((the carrier of X) --> x) by Def3
      .= X --> (h.x) by A1,FUNCOP_1:17;
  end;
  thus thesis;
end;
