theorem Th59:
  for f being PartFunc of [:X,Y:],Z for g being PartFunc of [:X9,Y9:],Z9
  holds |:f,g:| is PartFunc of [:[:X,X9:],[:Y,Y9:]:],[:Z,Z9:]
proof
  let f be PartFunc of [:X,Y:],Z;
  let g be PartFunc 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:];
  dom f c= [:X,Y:] & dom g c= [:X9,Y9:];
  then dom|:f,g:| c= [:[:X,X9:],[:Y,Y9:]:] by Th57;
  hence thesis by A1,RELSET_1:4;
end;
