reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th4:
  for S,T being complete LATTICE, f being sups-preserving Function
  of S,T st T is meet-continuous & f is meet-preserving one-to-one holds S is
  meet-continuous
proof
  let S,T be complete LATTICE, f be sups-preserving Function of S,T;
  assume that
A1: T is meet-continuous and
A2: f is meet-preserving one-to-one;
  S is satisfying_MC
  proof
    let x be Element of S, D be non empty directed Subset of S;
A3: ex_sup_of D,S & f preserves_sup_of D by WAYBEL_0:75,def 33;
    reconsider Y = {x} as directed non empty Subset of S by WAYBEL_0:5;
A4: ex_sup_of Y"/\"D,S & f preserves_sup_of {x}"/\"D by WAYBEL_0:75,def 33;
    reconsider X = f.:D as directed Subset of T by Lm1,YELLOW_2:15;
A5: dom f = the carrier of S by FUNCT_2:def 1;
A6: {f.x} "/\" (f.:D) = {(f.x) "/\" y where y is Element of T: y in (f.:D)
    } by YELLOW_4:42;
A7: {f.x} "/\" (f.:D) c= f.:({x}"/\"D)
    proof
      let p be object;
      assume p in {f.x} "/\" (f.:D);
      then consider y be Element of T such that
A8:   p = (f.x) "/\" y and
A9:   y in (f .:D) by A6;
      consider k be object such that
A10:  k in dom f and
A11:  k in D and
A12:  y = f.k by A9,FUNCT_1:def 6;
      reconsider k as Element of S by A10;
      (x "/\" k) in {x "/\" a where a is Element of S: a in D} by A11;
      then
A13:  (x "/\" k) in ({x} "/\" D) by YELLOW_4:42;
      f preserves_inf_of {x,k} by A2;
      then p = f.(x "/\" k) by A8,A12,YELLOW_3:8;
      hence thesis by A5,A13,FUNCT_1:def 6;
    end;
A14: {x} "/\" D = {x "/\" y where y is Element of S: y in D} by YELLOW_4:42;
A15: f.:({x}"/\"D) c= {f.x} "/\" (f.:D)
    proof
      let p be object;
      assume p in f.:({x}"/\"D);
      then consider m be object such that
A16:  m in dom f and
A17:  m in ({x} "/\" D) and
A18:  p = f.m by FUNCT_1:def 6;
      reconsider m as Element of S by A16;
      consider a be Element of S such that
A19:  m = x "/\" a and
A20:  a in D by A14,A17;
      reconsider fa = f.a as Element of T;
      f preserves_inf_of {x,a} by A2;
      then
A21:  p = (f.x) "/\" fa by A18,A19,YELLOW_3:8;
      fa in f.:D by A5,A20,FUNCT_1:def 6;
      hence thesis by A6,A21;
    end;
    f.(x "/\" sup D) = (f.x) "/\" (f.sup D) by A2,Th1
      .= (f.x) "/\" sup X by A3
      .= sup ({f.x} "/\" X) by A1,WAYBEL_2:def 6
      .= sup (f.:({x} "/\" D)) by A7,A15,XBOOLE_0:def 10
      .= f.(sup ({x} "/\" D)) by A4;
    hence x "/\" sup D = sup ({x} "/\" D) by A2;
  end;
  hence thesis;
end;
