reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem Th60:
  for f being Function of [:X,Y:],Z for g being Function of [:X9,
Y9:],Z9 st Z <> {} & Z9 <> {} holds |:f,g:| is Function of [:[:X,X9:],[:Y,Y9:]
  :],[:Z,Z9:]
proof
  let f be Function of [:X,Y:],Z;
  let g be Function of [:X9,Y9:],Z9;
  rng |:f,g:| c= [:rng f,rng g:] & [:rng f,rng g:] c= [:Z,Z9:] by Th56,
ZFMISC_1:96;
  then
A1: rng|:f,g:| c= [:Z,Z9:];
  assume
A2: Z <> {} & Z9 <> {};
  then dom f = [:X,Y:] & dom g = [:X9,Y9:] by FUNCT_2:def 1;
  then [:[:X,X9:],[:Y,Y9:]:] = dom|:f,g:| by Th58;
  then reconsider
  R = |:f,g:| as Relation of [:[:X,X9:],[:Y,Y9:]:],[:Z,Z9:] by A1,RELSET_1:4;
  R is quasi_total
  proof
    per cases;
    case [:Z,Z9:] <> {};
      dom f = [:X,Y:] & dom g = [:X9,Y9:] by A2,FUNCT_2:def 1;
      hence thesis by Th58;
    end;
    case [:Z,Z9:] = {};
      hence thesis;
    end;
  end;
  hence thesis;
end;
