
theorem
  for S, T being non empty RelStr st S, T are_isomorphic & S is
  with_infima holds T is with_infima
proof
  let S, T be non empty RelStr;
  given f being Function of S, T such that
A1: f is isomorphic;
  assume
A2: for a, b being Element of S ex c being Element of S st c <= a & c <=
  b & for c9 being Element of S st c9 <= a & c9 <= b holds c9 <= c;
  let x, y be Element of T;
  consider c being Element of S such that
A3: c <= f/".x & c <= f/".y and
A4: for c9 being Element of S st c9 <= f/".x & c9 <= f/".y holds c9 <= c by A2;
  take f.c;
A5: ex g being Function of T, S st g = f qua Function" & g is monotone by A1,
WAYBEL_0:def 38;
A6: rng f = the carrier of T by A1,WAYBEL_0:66;
A7: f/" = (f qua Function)" by A1,TOPS_2:def 4;
  f.c <= f.(f/".x) & f.c <= f.(f/".y) by A1,A3,WAYBEL_0:66;
  hence f.c <= x & f.c <= y by A1,A6,A7,FUNCT_1:35;
  let z9 be Element of T;
  assume z9 <= x & z9 <= y;
  then f/".z9 <= f/".x & f/".z9 <= f/".y by A7,A5,WAYBEL_1:def 2;
  then f/".z9 <= c by A4;
  then f.(f/".z9) <= f.c by A1,WAYBEL_0:66;
  hence thesis by A1,A6,A7,FUNCT_1:35;
end;
