theorem
  (F*(id D,u)).:(T1,T2) = F.:(T1,u*T2)
proof
  now
    per cases;
    suppose
      i = 0;
      then (F*(id D,u)).:(T1,T2) = <*>D & u*T2 = <*>D by Lm1;
      hence thesis by FINSEQ_2:73;
    end;
    suppose
      i <> 0;
      then reconsider C = Seg i as non empty set;
      T1 is Function of C,D & T2 is Function of C,D by Lm4;
      hence thesis by Th82;
    end;
  end;
  hence thesis;
end;
