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;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;
reserve p9,q9 for Element of LattPOSet L;
reserve C for complete Lattice,
  a,a9,b,b9,c,d for Element of C,
  X,Y for set;

theorem Th46:
  "\/"(X,C) = "\/"({a: ex b st a [= b & b in X}, C) &
  "/\"(X,C) = "/\"({b: ex a st a [= b & a in X}, C)
proof
  set Y = {a: ex b st a [= b & b in X}, Z = {a: ex b st b [= a & b in X};
  X is_less_than "\/"(Y,C)
  proof
    let a;
    assume a in X;
    then a in Y;
    hence thesis by Th38;
  end;
  then
A1: "\/"(X,C) [= "\/"(Y,C) by Def21;
  Y is_less_than "\/"(X,C)
  proof
    let a;
    assume a in Y;
    then ex b st a = b & ex c st b [= c & c in X;
    then consider c such that
A2: a [= c and
A3: c in X;
    c [= "\/"(X,C) by A3,Th38;
    hence thesis by A2,LATTICES:7;
  end;
  then "\/"(Y,C) [= "\/"(X,C) by Def21;
  hence "\/"(X,C) = "\/"(Y,C) by A1,LATTICES:8;
  X is_greater_than "/\"(Z,C)
  proof
    let a;
    assume a in X;
    then a in Z;
    hence thesis by Th38;
  end;
  then
A4: "/\"(Z,C) [= "/\"(X,C) by Th34;
  Z is_greater_than "/\"(X,C)
  proof
    let a;
    assume a in Z;
    then ex b st a = b & ex c st c [= b & c in X;
    then consider c such that
A5: c [= a and
A6: c in X;
    "/\"(X,C) [= c by A6,Th38;
    hence thesis by A5,LATTICES:7;
  end;
  then "/\"(X,C) [= "/\"(Z,C) by Th34;
  hence thesis by A4,LATTICES:8;
end;
