theorem
  F is_distributive_wrt G implies
    F[:](T,G.(d1,d2)) = G.:(F[:](T,d1),F [:] (T,d2))
proof
  assume
A1: F is_distributive_wrt G;
  per cases;
  suppose
    i = 0;
    then F[:](T,d1) = <*>D & F[:](T,G.(d1,d2)) = <*>D by Lm3;
    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,Th36;
  end;
end;
