reserve X1, X2, Y for non empty RelStr,
  f for Function of [:X1, X2:], Y,
  x for Element of X1,
  y for Element of X2;

theorem Th14:
  for R, S, T being LATTICE, f being Function of [:R,S:], T, b
  being Element of S, X being Subset of R holds Proj (f, b).:X = f.:[:X, {b}:]
proof
  let R, S, T be LATTICE, f be Function of [:R,S:], T, b be Element of S, X be
  Subset of R;
A1: Proj (f, b).:X c= f.:[:X, {b}:]
  proof
    let c be object;
    assume c in Proj (f, b).:X;
    then consider k be object such that
A2: k in dom Proj (f, b) and
A3: k in X and
A4: c = Proj (f, b).k by FUNCT_1:def 6;
    b in {b} by TARSKI:def 1;
    then
A5: [k, b] in [:X, {b}:] by A3,ZFMISC_1:87;
    [:the carrier of R, the carrier of S:] = the carrier of [:R, S:] by
YELLOW_3:def 2;
    then dom f = [:the carrier of R, the carrier of S:] by FUNCT_2:def 1;
    then
A6: [k, b] in dom f by A2,ZFMISC_1:87;
    c = f.(k, b) by A2,A4,Th8;
    hence thesis by A5,A6,FUNCT_1:def 6;
  end;
  f.:[:X, {b}:] c= Proj (f, b).:X
  proof
    let c be object;
    assume c in f.:[:X, {b}:];
    then consider k be object such that
    k in dom f and
A7: k in [:X, {b}:] and
A8: c = f.k by FUNCT_1:def 6;
    consider k1, k2 be object such that
A9: k1 in X and
A10: k2 in {b} and
A11: k = [k1, k2] by A7,ZFMISC_1:def 2;
A12: k2 = b by A10,TARSKI:def 1;
    c = f.(k1,k2) by A8,A11;
    then dom Proj (f, b) = the carrier of R & c = Proj (f, b). k1 by A9,A12,Th8
,FUNCT_2:def 1;
    hence thesis by A9,FUNCT_1:def 6;
  end;
  hence thesis by A1;
end;
