reserve i,j,k,x for object;

theorem Th5:
  for A,B being functional set for F being compositional
ManySortedSet of [:A,B:], g,f being Function st g in A & f in B holds F.(g,f) =
  g*f
proof
  let A,B be functional set;
  let F be compositional ManySortedSet of [:A,B:], g,f be Function such that
A1: g in A & f in B;
  dom F = [:A,B:] by PARTFUN1:def 2;
  then [g,f] in dom F by A1,ZFMISC_1:87;
  then consider ff,gg being Function such that
A2: [g,f] = [gg,ff] and
A3: F.[g,f] = gg*ff by Def9;
  g = gg by A2,XTUPLE_0:1;
  hence thesis by A2,A3,XTUPLE_0:1;
end;
