theorem
  F is_distributive_wrt G implies
    F[;](G.(d1,d2),T) = G.:(F[;](d1,T),F[;](d2,T))
proof
  assume
A1: F is_distributive_wrt G;
  per cases;
  suppose
    i = 0;
    then F[;](d1,T) = <*>D & F[;](G.(d1,d2),T) = <*>D by Lm2;
    hence thesis by FINSEQ_2:73;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
    T is Function of C,D by Lm4;
    hence thesis by A1,Th35;
  end;
end;
