theorem Th77:
  for F,f,g being Function st [x,y] in dom(F*(f,g)) holds
    (F*(f,g)).(x,y) = F.(f.x,g.y)
proof
  let F,f,g be Function such that
A1: [x,y] in dom(F*(f,g));
  [x,y] in dom [:f,g:] by A1,FUNCT_1:11;
  then [x,y] in [:dom f,dom g:] by FUNCT_3:def 8;
  then [:f,g:].(x,y) = [f.x,g.y] by FUNCT_3:65;
  hence thesis by A1,FUNCT_1:12;
