
theorem
  for S, T being non empty RelStr st S, T are_isomorphic & S is
  with_suprema holds T is with_suprema
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 a <= c & b <=
  c & for c9 being Element of S st a <= c9 & b <= c9 holds c <= c9;
  let x, y be Element of T;
  consider c being Element of S such that
A3: f/".x <= c & f/".y <= c and
A4: for c9 being Element of S st f/".x <= c9 & f/".y <= c9 holds c <= c9 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.(f/".x) <= f.c & f.(f/".y) <= f.c by A1,A3,WAYBEL_0:66;
  hence x <= f.c & y <= f.c by A1,A6,A7,FUNCT_1:35;
  let z9 be Element of T;
  assume x <= z9 & y <= z9;
  then f/".x <= f/".z9 & f/".y <= f/".z9 by A7,A5,WAYBEL_1:def 2;
  then c <= f/".z9 by A4;
  then f.c <= f.(f/".z9) by A1,WAYBEL_0:66;
  hence thesis by A1,A6,A7,FUNCT_1:35;
end;
