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 Th10:
  for R, S, T being non empty reflexive RelStr, f being Function
  of [:R,S:], T, a being Element of R, b being Element of S st f is monotone
  holds Proj (f, a) is monotone & Proj (f, b) is monotone
proof
  let R, S, T be non empty reflexive RelStr, f be Function of [:R,S:], T;
  let a be Element of R, b be Element of S;
  reconsider a as Element of R;
  reconsider b as Element of S;
  set g = Proj (f, b), h = Proj (f, a);
  assume
A1: f is monotone;
A2: now
    let x, y be Element of R;
A3: b <= b;
A4: g. x = f.(x, b) & g. y = f.(y, b) by Th8;
    assume x <= y;
    then [x, b] <= [y, b] by A3,YELLOW_3:11;
    hence g.x <= g.y by A1,A4,WAYBEL_1:def 2;
  end;
  now
    let x, y be Element of S;
A5: a <= a;
A6: h. x = f.(a, x) & h. y = f.(a, y) by Th7;
    assume x <= y;
    then [a, x] <= [a, y] by A5,YELLOW_3:11;
    hence h.x <= h.y by A1,A6,WAYBEL_1:def 2;
  end;
  hence thesis by A2,WAYBEL_1:def 2;
end;
