reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;
reserve S for non empty TopStruct;
reserve f for Function of T,S;

theorem
  T is compact & f is continuous & rng f = [#] S implies S is compact
proof
  assume
A1: T is compact;
  [#] T c= dom f by FUNCT_2:def 1;
  then
A2: [#] T c= f"(f.:([#] T)) by FUNCT_1:76;
  assume that
A3: f is continuous and
A4: rng f = [#] S;
  let F be Subset-Family of S such that
A5: F is Cover of S and
A6: F is open;
  set F1=F;
  reconsider G = ("f).:F1 as Subset-Family of T by TOPS_2:42;
  union G = f"(union F) by A4,FUNCT_3:26
    .= f"(rng f) by A4,A5,SETFAM_1:45
    .= f"(f.: (dom f)) by RELAT_1:113
    .= f"(f.:([#] T)) by FUNCT_2:def 1;
  then
A7: G is Cover of T by A2,SETFAM_1:def 11;
A8: (.:f).:((.:f)"F) c= F by FUNCT_1:75;
  (.:f).:(("f).:F) c= (.:f).:((.:f)"F) by FUNCT_3:29,RELAT_1:123;
  then
A9: (.:f).:G c= F by A8;
  G is open by A3,A6,TOPS_2:47;
  then consider G9 being Subset-Family of T such that
A10: G9 c= G and
A11: G9 is Cover of T and
A12: G9 is finite by A1,A7;
  reconsider F9=(.:f).:G9 as Subset-Family of S;
  take F9;
  (.:f).:G9 c= (.:f).:G by A10,RELAT_1:123;
  hence F9 c= F by A9;
  dom f = [#] T by FUNCT_2:def 1;
  then union F9 = f.:(union G9) by FUNCT_3:14
    .= f.:([#] T) by A11,SETFAM_1:45
    .= f.:dom f by FUNCT_2:def 1
    .= [#] S by A4,RELAT_1:113;
  hence F9 is Cover of S by SETFAM_1:def 11;
  thus thesis by A12;
end;
