theorem Th12:
  for C being non empty set, f being PartFunc of C,ExtREAL,
  x being Element of C holds 0. <= (max+(f)).x
proof
  let C be non empty set;
  let f be PartFunc of C,ExtREAL;
  let x be Element of C;
A1: dom max+f = dom f by Def2;
  per cases;
  suppose
 x in dom f;
then  (max+(f).x) = max(f.x,0.) by A1,Def2;
    hence thesis by XXREAL_0:25;
  end;
  suppose
 not x in dom f;
    hence thesis by A1,FUNCT_1:def 2;
  end;
end;
