reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th26:
  for X being finite non empty Subset of REAL, f being Function of
[:X,X:],REAL, z being finite non empty Subset of REAL, A being Real st z
  = rng f & A >= max z holds for x,y being Element of X holds f.(x,y) <= A
proof
  let X be finite non empty Subset of REAL, f be Function of [:X,X:],REAL, z
  be finite non empty Subset of REAL, A be Real such that
A1: z = rng f and
A2: A >= max z;
  now
    let x,y be Element of X;
    reconsider c = f.[x,y] as Real;
    dom f = [:X,X:] by FUNCT_2:def 1;
    then [x,y] in dom f by ZFMISC_1:def 2;
    then c in z by A1,FUNCT_1:def 3;
    then f.(x,y) <= max z by XXREAL_2:def 8;
    hence f.(x,y) <= A by A2,XXREAL_0:2;
  end;
  hence thesis;
end;
