
theorem
  for S, T being non empty RelStr, A being Subset of S, f being Function
  of S, T st f is isomorphic & ex_inf_of A, S holds ex_inf_of f.:A, T
proof
  let S, T be non empty RelStr, A be Subset of S, f be Function of S, T such
  that
A1: f is isomorphic;
A2: f"(f.:A) c= A by A1,FUNCT_1:82;
A3: rng f = [#]T by A1,WAYBEL_0:66;
A4: f/" = f qua Function " by A1,TOPS_2:def 4;
  given a being Element of S such that
A5: A is_>=_than a and
A6: for b being Element of S st A is_>=_than b holds b <= a and
A7: for c being Element of S st A is_>=_than c & for b being Element of
  S st A is_>=_than b holds b <= c holds c = a;
  take f.a;
  thus f.:A is_>=_than f.a by A1,A5,WAYBEL13:18;
A8: f/" is isomorphic by A1,Th10;
A9: dom f = the carrier of S by FUNCT_2:def 1;
  then A c= f"(f.:A) by FUNCT_1:76;
  then
A10: f"(f.:A) = A by A2;
  hereby
    let b be Element of T;
    assume f.:A is_>=_than b;
    then f/".:(f.:A) is_>=_than f/".b by A8,WAYBEL13:18;
    then f"(f.:A) is_>=_than f/".b by A1,A3,TOPS_2:55;
    then f/".b <= a by A6,A10;
    then f.(f/".b) <= f.a by A1,WAYBEL_0:66;
    hence b <= f.a by A1,A3,A4,FUNCT_1:35;
  end;
  let c be Element of T;
  assume
A11: f.:A is_>=_than c;
  assume
A12: for b being Element of T st f.:A is_>=_than b holds b <= c;
A13: for b being Element of S st A is_>=_than b holds b <= f/".c
  proof
    let b be Element of S;
    assume A is_>=_than b;
    then f.:A is_>=_than f.b by A1,WAYBEL13:18;
    then f.b <= c by A12;
    then f/".(f.b) <= f/".c by A8,WAYBEL_0:66;
    hence thesis by A1,A4,A9,FUNCT_1:34;
  end;
  f/".:(f.:A) is_>=_than f/".c by A8,A11,WAYBEL13:18;
  then A is_>=_than f/".c by A1,A3,A10,TOPS_2:55;
  then f/".c = a by A7,A13;
  hence thesis by A1,A3,A4,FUNCT_1:35;
end;
