reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;

theorem Th3:
  H1 = G - H iff for n holds H1.n = G.n - H.n
proof
  thus H1 = G - H implies for n holds H1.n = G.n - H.n
  proof
    assume
A1: H1 = G - H;
    let n;
    thus H1.n = G.n + (-H).n by A1,Def5
      .= G.n - H.n by Def3;
  end;
  assume
A2: for n holds H1.n = G.n - H.n;
  now
    let n be Element of NAT;
    thus H1.n = G.n - H.n by A2
      .= G.n + (-H).n by Def3
      .= (G - H).n by Def5;
  end;
  hence thesis by FUNCT_2:63;
end;
