theorem Th13:
  (for l being Nat of n st l=i holds (x+*(i,v)).l = v) & for l,i
  being Nat of n st l<>i holds (x+*(i,v)).l = x.l
proof
  thus for l being Nat of n st l=i holds (x+*(i,v)).l = v
  proof
    let l be Nat of n such that
A1: l = i;
    l in Seg(n+1) by Th7;
    hence thesis by A1,Lm1;
  end;
  thus thesis by FUNCT_7:32;
end;
