
theorem Th14:
  for T being Semilattice for S being full non empty SubRelStr of T
  holds S is meet-inheriting iff
  for X being finite non empty Subset of S holds "/\"
  (X, T) in the carrier of S
proof
  let T be Semilattice;
  let S be full non empty SubRelStr of T;
  hereby
    assume
A1: S is meet-inheriting;
    let X be finite non empty Subset of S;
A2: X is finite;
    defpred P[set] means $1 <> {} implies "/\"($1, T) in the carrier of S;
A3: P[{}];
A4: now
      let y,x be set;
      assume that
A5:   y in X and
A6:   x c= X and
A7:   P[x];
      thus P[x \/ {y}]
      proof
        assume x \/ {y} <> {};
        reconsider y9 = y as Element of S by A5;
        reconsider z = y9 as Element of T by YELLOW_0:58;
A8:     x c= the carrier of S by A6,XBOOLE_1:1;
        the carrier of S c= the carrier of T by YELLOW_0:def 13;
        then reconsider x9 = x as finite Subset of T
        by A6,A8,XBOOLE_1:1;
A9:     ex_inf_of {z}, T by YELLOW_0:38;
A10:    inf {z} = y9 by YELLOW_0:39;
        now
          assume
A11:      x9 <> {};
          then ex_inf_of x9, T by YELLOW_0:55;
          then
A12:      inf (x9 \/ {z}) = (inf x9 )"/\"z by A9,A10,YELLOW_2:4;
          ex_inf_of {inf x9, z}, T by YELLOW_0:21;
          then inf {inf x9, z} in the carrier of S by A1,A7,A11;
          hence inf (x9 \/ {z}) in the carrier of S by A12,YELLOW_0:40;
        end;
        hence thesis by A10;
      end;
    end;
    P[X] from FINSET_1:sch 2(A2,A3,A4);
    hence "/\"(X, T) in the carrier of S;
  end;
  assume
A13: for X being finite non empty Subset of S holds "/\"
  (X, T) in the carrier of S;
  let x,y be Element of T;
  assume that
A14: x in the carrier of S and
A15: y in the carrier of S;
  {x,y} c= the carrier of S by A14,A15,ZFMISC_1:32;
  hence thesis by A13;
end;
