reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;

theorem Th12:
  A is complete implies A is with_suprema with_infima
proof
  assume
A1: for X being set ex a st X is_<=_than a &
  for b st X is_<=_than b holds a <= b;
  thus A is with_suprema
  proof
    let a,b;
    consider c such that
A2: {a,b} is_<=_than c and
A3: for c9 st {a,b} is_<=_than c9 holds c <= c9 by A1;
    take c;
A4: a in {a,b} by TARSKI:def 2;
    b in {a,b} by TARSKI:def 2;
    hence a <= c & b <= c by A2,A4;
    let c9 such that
A5: a <= c9 and
A6: b <= c9;
    {a,b} is_<=_than c9
    by A5,A6,TARSKI:def 2;
    hence thesis by A3;
  end;
  let a,b;
  set X = {c: c <= a & c <= b};
  consider c such that
A7: X is_<=_than c and
A8: for c9 st X is_<=_than c9 holds c <= c9 by A1;
  take c;
  X is_<=_than a
  proof
    let c;
    assume c in X;
    then ex c9 st c = c9 & c9 <= a & c9 <= b;
    hence thesis;
  end;
  hence c <= a by A8;
  X is_<=_than b
  proof
    let c;
    assume c in X;
    then ex c9 st c = c9 & c9 <= a & c9 <= b;
    hence thesis;
  end;
  hence c <= b by A8;
  let c9;
  assume that
A9: c9 <= a and
A10: c9 <= b;
  c9 in X by A9,A10;
  hence thesis by A7;
end;
