reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,i for set;
reserve r,r1,r2 for Real;

theorem Th11:
  for L be Lattice, L9 be SubLattice of L, a,b be Element of L, a9
,b9 be Element of L9 st a = a9 & b = b9 holds a "\/" b = a9 "\/" b9 & a "/\" b
  = a9 "/\" b9
proof
  let L be Lattice;
  let L9 be SubLattice of L;
  let a,b be Element of L;
  let a9,b9 be Element of L9;
  assume
A1: a = a9 & b = b9;
  thus a "\/" b = (the L_join of L).(a,b) by LATTICES:def 1
    .= ((the L_join of L)||the carrier of L9). [a9,b9] by A1,FUNCT_1:49
    .= (the L_join of L9).(a9,b9) by NAT_LAT:def 12
    .= a9 "\/" b9 by LATTICES:def 1;
  thus a "/\" b = (the L_meet of L).(a,b) by LATTICES:def 2
    .= ((the L_meet of L)||the carrier of L9). [a9,b9] by A1,FUNCT_1:49
    .= (the L_meet of L9).(a9,b9) by NAT_LAT:def 12
    .= a9 "/\" b9 by LATTICES:def 2;
end;
