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 Th17:
  for R, S, T being LATTICE, f being monotone Function of [:R, S:]
, T, a being Element of R, b being Element of S, X being directed Subset of [:R
, S:] st ex_sup_of f.:X, T & a in proj1 X & b in proj2 X holds f. [a, b] <= sup
  (f.:X)
proof
  let R, S, T be LATTICE, f be monotone Function of [:R, S:], T, a be Element
  of R, b be Element of S, X be directed Subset of [:R, S:];
  assume that
A1: ex_sup_of f.:X, T and
A2: a in proj1 X and
A3: b in proj2 X;
  consider d being object such that
A4: [d, b] in X by A3,XTUPLE_0:def 13;
  d in proj1 X by A4,XTUPLE_0:def 12;
  then reconsider d as Element of R;
  consider c being object such that
A5: [a, c] in X by A2,XTUPLE_0:def 12;
  c in proj2 X by A5,XTUPLE_0:def 13;
  then reconsider c as Element of S;
  consider z being Element of [:R, S:] such that
A6: z in X and
A7: [a, c] <= z & [d, b] <= z by A5,A4,WAYBEL_0:def 1;
A8: f.:X is_<=_than sup (f.:X) by A1,YELLOW_0:30;
  [a, c] "\/" [d, b] <= z "\/" z by A7,YELLOW_3:3;
  then [a, c] "\/" [d, b] <= z by YELLOW_5:1;
  then [a "\/" d, c "\/" b] <= z by YELLOW10:16;
  then
A9: f. [a "\/" d, c "\/" b] <= f. z by WAYBEL_1:def 2;
  dom f = the carrier of [:R, S:] by FUNCT_2:def 1;
  then f.z in f.:X by A6,FUNCT_1:def 6;
  then
A10: f. z <= sup (f.:X) by A8;
  a <= a "\/" d & b <= c "\/" b by YELLOW_0:22;
  then [a, b] <= [a "\/" d, c "\/" b] by YELLOW_3:11;
  then f. [a, b] <= f. [a "\/" d, c "\/" b] by WAYBEL_1:def 2;
  then f. [a, b] <= f. z by A9,YELLOW_0:def 2;
  hence thesis by A10,YELLOW_0:def 2;
end;
