
theorem Th29:
  for L being Semilattice, A being Subset of L for B being non
  empty Subset of L st A is_coarser_than B holds fininfs A is_coarser_than
  fininfs B
proof
  let L be Semilattice, A be Subset of L, B be non empty Subset of L such that
A1: for a being Element of L st a in A ex b being Element of L st b in B
  & b <= a;
  defpred P[object,object] means
    ex x, y being Element of L st x = $1 & y = $2 & y <= x;
  let a be Element of L;
  assume a in fininfs A;
  then consider Y being finite Subset of A such that
A2: a = "/\"(Y,L) and
A3: ex_inf_of Y,L;
A4: for e being object st e in Y ex u being object st u in B & P[e,u]
  proof
    let e be object such that
A5: e in Y;
    Y c= the carrier of L by XBOOLE_1:1;
    then reconsider e as Element of L by A5;
    ex b being Element of L st b in B & b <= e by A1,A5;
    hence thesis;
  end;
  consider f being Function of Y, B such that
A6: for e being object st e in Y holds P[e,f.e] from FUNCT_2:sch 1(A4);
A7: f.:Y c= the carrier of L
  proof
    let y be object;
    assume y in f.:Y;
    then consider x being object such that
A8: x in dom f and
    x in Y and
A9: y = f.x by FUNCT_1:def 6;
    f.x in B by A8,FUNCT_2:5;
    hence thesis by A9;
  end;
A10: now
    per cases;
    case
      Y = {};
      hence ex_inf_of f.:Y,L by A3;
    end;
    case
      Y <> {};
      then consider e being object such that
A11:  e in Y by XBOOLE_0:def 1;
      dom f = Y by FUNCT_2:def 1;
      then f.e in f.:Y by A11,FUNCT_1:def 6;
      hence ex_inf_of f.:Y,L by A7,YELLOW_0:55;
    end;
  end;
  "/\"(f.:Y,L) is_<=_than Y
  proof
    let e be Element of L;
    assume
A12: e in Y;
    then consider x, y being Element of L such that
A13: x = e and
A14: y = f.e and
A15: y <= x by A6;
    dom f = Y by FUNCT_2:def 1;
    then f.e in f.:Y by A12,FUNCT_1:def 6;
    then "/\"(f.:Y,L) <= y by A10,A14,YELLOW_4:2;
    hence "/\"(f.:Y,L) <= e by A13,A15,ORDERS_2:3;
  end;
  then
A16: "/\"(f.:Y,L) <= "/\"(Y,L) by A3,YELLOW_0:31;
  "/\"(f.:Y,L) in fininfs B by A10;
  hence ex b being Element of L st b in fininfs B & b <= a by A2,A16;
end;
