reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th26:
  for L being non empty RelStr holds (for A being Subset of L
  holds ex_inf_of A, L) implies for X holds ex_inf_of X,L & "/\"(X, L) = "/\"(X
  /\ the carrier of L, L)
proof
  let L be non empty RelStr;
  assume
A1: for A being Subset of L holds ex_inf_of A, L;
  let X;
  set Y = X /\ the carrier of L;
  set a = "/\"(Y, L);
  reconsider Y as Subset of L by XBOOLE_1:17;
A2: ex_inf_of Y,L by A1;
A3: a is_<=_than X
  proof
    let x be Element of L;
    assume x in X;
    then
A4: x in Y by XBOOLE_0:def 4;
    a is_<=_than Y by A2,YELLOW_0:def 10;
    hence thesis by A4;
  end;
A5: for b being Element of L st b is_<=_than X holds b <= a
  proof
    let b be Element of L;
A6: Y c= X by XBOOLE_1:17;
    assume b is_<=_than X;
    then b is_<=_than Y by A6;
    hence thesis by A2,YELLOW_0:def 10;
  end;
  ex_inf_of X,L by A2,YELLOW_0:50;
  hence thesis by A3,A5,YELLOW_0:def 10;
end;
