
theorem Th2:
  for S,T being Semilattice, f being Function of S,T st f is meet-preserving
  for X being finite non empty Subset of S holds f preserves_inf_of X
proof
  let S,T be Semilattice, f be Function of S,T such that
A1: f is meet-preserving;
  let X be finite non empty Subset of S such that ex_inf_of X,S;
A2: X is finite;
  defpred P[set] means $1 <> {} implies ex_inf_of $1, S & ex_inf_of f.:$1, T &
  inf (f.:$1) = f."/\"($1,S);
A3: P[{}];
A4: now
    let y,x be set such that
A5: y in X and x c= X and
A6: P[x];
    thus P[x \/ {y}]
    proof
      assume x \/ {y} <> {};
      reconsider y9 = y as Element of S by A5;
      set fy = f.y9;
A7:   ex_inf_of {fy}, T by YELLOW_0:38;
A8:   inf {fy} = fy by YELLOW_0:39;
A9:   ex_inf_of {y9}, S by YELLOW_0:38;
A10:  inf {y9} = y by YELLOW_0:39;
      thus ex_inf_of x \/ {y}, S by A6,A9,YELLOW_2:4;
      dom f = the carrier of S by FUNCT_2:def 1;
      then
A11:  Im(f,y) = {fy} by FUNCT_1:59;
      then
A12:  f.:(x \/ {y}) = (f.:x) \/ {fy} by RELAT_1:120;
      hence ex_inf_of f.:(x \/ {y}), T by A6,A7,A11,YELLOW_2:4;
      per cases;
      suppose x = {};
        hence thesis by A8,A11,YELLOW_0:39;
      end;
      suppose
A13:    x <> {};
        hence "/\"(f.:(x \/ {y}), T)
        = (f."/\"(x, S)) "/\" fy by A6,A7,A8,A12,YELLOW_2:4
          .= f.("/\"(x, S)"/\" y9) by A1,WAYBEL_6:1
          .= f."/\"(x \/ {y}, S) by A6,A9,A10,A13,YELLOW_2:4;
      end;
    end;
  end;
  P[X] from FINSET_1:sch 2(A2,A3,A4);
  hence thesis;
end;
