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 Th15:
  for R, S, T being LATTICE, f being Function of [:R,S:], T, b
  being Element of R, X being Subset of S holds Proj (f, b).:X = f.:[:{b}, X:]
proof
  let R, S, T be LATTICE, f be Function of [:R,S:], T, b be Element of R, X be
  Subset of S;
A1: Proj (f, b).:X c= f.:[:{b}, X:]
  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: [b, k] in [:{b}, X:] 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: [b, k] in dom f by A2,ZFMISC_1:87;
    c = f.(b, k) by A2,A4,Th7;
    hence thesis by A5,A6,FUNCT_1:def 6;
  end;
  f.:[:{b}, X:] c= Proj (f, b).:X
  proof
    let c be object;
    assume c in f.:[:{b}, X:];
    then consider k be object such that
    k in dom f and
A7: k in [:{b}, X:] and
A8: c = f.k by FUNCT_1:def 6;
    consider k1, k2 be object such that
A9: k1 in {b} and
A10: k2 in X and
A11: k = [k1, k2] by A7,ZFMISC_1:def 2;
A12: k1 = b by A9,TARSKI:def 1;
    c = f.(k1,k2) by A8,A11;
    then dom Proj (f, b) = the carrier of S & c = Proj (f, b). k2 by A10,A12
,Th7,FUNCT_2:def 1;
    hence thesis by A10,FUNCT_1:def 6;
  end;
  hence thesis by A1;
end;
