reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

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;
