reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;

theorem Th45:
  for T being TopSpace, S being non empty TopSpace, f being
Function of T, S holds f is continuous iff for P being Subset of T holds f.:(Cl
  P) c= Cl(f.:P)
proof
  let T be TopSpace, S be non empty TopSpace, f be Function of T, S;
  hereby
    assume
A1: f is continuous;
    let P be Subset of T;
    P c= [#]T;
    then P c= dom f by FUNCT_2:def 1;
    then
A2: Cl P c= Cl(f"(f.:P)) by FUNCT_1:76,PRE_TOPC:19;
    Cl(f"(f.:P)) c= f"(Cl(f.:P)) by A1,Th44;
    then Cl P c= f"(Cl(f.:P)) by A2;
    then
A3: f.:(Cl P) c= f.:(f"(Cl(f.:P))) by RELAT_1:123;
    f.:(f"(Cl(f.:P))) c= Cl(f.:P) by FUNCT_1:75;
    hence f.:(Cl P) c= Cl(f.:P) by A3;
  end;
  assume
A4: for P being Subset of T holds f.:(Cl P) c= Cl(f.:P);
  now
    let P1 be Subset of S;
    Cl(f"P1) c= [#]T;
    then Cl(f"P1) c= dom f by FUNCT_2:def 1;
    then
A5: Cl(f"P1) c= f"(f.:Cl(f"P1)) by FUNCT_1:76;
    f.:(Cl(f"P1)) c= Cl(f.:(f"P1)) & Cl(f.:(f"P1)) c= Cl P1 by A4,FUNCT_1:75
,PRE_TOPC:19;
    then f.:(Cl(f"P1)) c= Cl P1;
    then f"(f.:(Cl(f"P1))) c= f"(Cl P1) by RELAT_1:143;
    hence Cl(f"P1) c= f"(Cl P1) by A5;
  end;
  hence thesis by Th44;
end;
