
theorem Th20:
  for L1,L2 be non empty antisymmetric RelStr for f be Function of
  L1,L2 holds f is isomorphic implies f is infs-preserving sups-preserving
proof
  let L1,L2 be non empty antisymmetric RelStr;
  let f be Function of L1,L2;
  assume
A1: f is isomorphic;
  then
A2: rng f = the carrier of L2 by WAYBEL_0:66;
  now
    let X be Subset of L1;
    now
      assume
A3:   ex_inf_of X,L1;
      then consider a be Element of L1 such that
A4:   X is_>=_than a and
A5:   for b be Element of L1 st X is_>=_than b holds a >= b by YELLOW_0:16;
A6:   now
        let c be Element of L2;
        consider e be object such that
A7:     e in dom f and
A8:     c = f.e by A2,FUNCT_1:def 3;
        reconsider e as Element of L1 by A7;
        assume f.:X is_>=_than c;
        then X is_>=_than e by A1,A8,Th18;
        then a >= e by A5;
        hence f.a >= c by A1,A8,WAYBEL_0:66;
      end;
      f.:X is_>=_than f.a by A1,A4,Th18;
      hence ex_inf_of f.:X,L2 by A6,YELLOW_0:16;
A9:   now
        let b be Element of L2;
        consider v be object such that
A10:    v in dom f and
A11:    b = f.v by A2,FUNCT_1:def 3;
        reconsider v as Element of L1 by A10;
        assume b is_<=_than f.:X;
        then v is_<=_than X by A1,A11,Th18;
        then inf X >= v by A3,YELLOW_0:31;
        hence f.inf X >= b by A1,A11,WAYBEL_0:66;
      end;
      inf X is_<=_than X by A3,YELLOW_0:31;
      then f.inf X is_<=_than f.:X by A1,Th18;
      hence inf (f.:X) = f.inf X by A9,YELLOW_0:31;
    end;
    hence f preserves_inf_of X by WAYBEL_0:def 30;
  end;
  hence f is infs-preserving by WAYBEL_0:def 32;
  now
    let X be Subset of L1;
    now
      assume
A12:  ex_sup_of X,L1;
      then consider a be Element of L1 such that
A13:  X is_<=_than a and
A14:  for b be Element of L1 st X is_<=_than b holds a <= b by YELLOW_0:15;
A15:  now
        let c be Element of L2;
        consider e be object such that
A16:    e in dom f and
A17:    c = f.e by A2,FUNCT_1:def 3;
        reconsider e as Element of L1 by A16;
        assume f.:X is_<=_than c;
        then X is_<=_than e by A1,A17,Th19;
        then a <= e by A14;
        hence f.a <= c by A1,A17,WAYBEL_0:66;
      end;
      f.:X is_<=_than f.a by A1,A13,Th19;
      hence ex_sup_of f.:X,L2 by A15,YELLOW_0:15;
A18:  now
        let b be Element of L2;
        consider v be object such that
A19:    v in dom f and
A20:    b = f.v by A2,FUNCT_1:def 3;
        reconsider v as Element of L1 by A19;
        assume b is_>=_than f.:X;
        then v is_>=_than X by A1,A20,Th19;
        then sup X <= v by A12,YELLOW_0:30;
        hence f.sup X <= b by A1,A20,WAYBEL_0:66;
      end;
      sup X is_>=_than X by A12,YELLOW_0:30;
      then f.sup X is_>=_than f.:X by A1,Th19;
      hence sup (f.:X) = f.sup X by A18,YELLOW_0:30;
    end;
    hence f preserves_sup_of X by WAYBEL_0:def 31;
  end;
  hence thesis by WAYBEL_0:def 33;
end;
