
theorem Th64:
  for S,T being LATTICE, f being join-preserving Function of S,T
  for S9 being non empty join-inheriting full SubRelStr of S
  for T9 being non empty join-inheriting full SubRelStr of T
  for g being Function of S9, T9 st g = f|the carrier of S9
  holds g is join-preserving
proof
  let S,T be LATTICE, f be join-preserving Function of S,T;
  let S9 be non empty join-inheriting full SubRelStr of S;
  let T9 be non empty join-inheriting full SubRelStr of T;
  let g be Function of S9, T9 such that
A1: g = f|the carrier of S9;
  now
    let x,y be Element of S9;
    reconsider a = x, b = y as Element of S by YELLOW_0:58;
    x"\/"y = a"\/"b by YELLOW_0:70;
    then
A2: g.(x"\/"y) = f.(a"\/"b) by A1,FUNCT_1:49;
A3: g.x = f.a by A1,FUNCT_1:49;
A4: g.y = f.b by A1,FUNCT_1:49;
    thus g.(x"\/"y) = (f.a)"\/"(f.b) by A2,WAYBEL_6:2
      .= (g.x)"\/"(g.y) by A3,A4,YELLOW_0:70;
  end;
  hence thesis by WAYBEL_6:2;
end;
