reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;

theorem Th29:
  for L being right_zeroed non empty addLoopStr, x,y being Element of L holds
  seq(n,x) + seq(n,y) = seq(n,x+y)
  proof
    let L be right_zeroed non empty addLoopStr, x,y be Element of L;
    let a be Element of NAT;
A1: (seq(n,x) + seq(n,y)).a = seq(n,x).a + seq(n,y).a by NORMSP_1:def 2;
    per cases;
    suppose a = n;
      then seq(n,x).a = x & seq(n,y).a = y & seq(n,x+y).a = x+y by Th24;
      hence thesis by NORMSP_1:def 2;
    end;
    suppose a <> n;
      then seq(n,x).a = 0.L & seq(n,y).a = 0.L & seq(n,x+y).a = 0.L by Th25;
      hence thesis by A1,RLVECT_1:def 4;
    end;
  end;
