theorem Th3:
  (for n holds a.n = 2*n + 1) implies a is increasing sequence of NAT
proof
  assume
A1: for n holds a.n = 2*n + 1;
A2: a is increasing
  proof
    let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
A4: 2*n + 1+0 < 2*n + 1+2 by XREAL_1:8;
    2*n + 1+2 = 2*(n + 1) + 1 .= a.(n + 1) by A1;
    hence a.n < a.(n+1) by A1,A4,A3;
  end;
A5: now
    let x;
    assume x in dom a;
    then reconsider n = x as Element of NAT;
    a.n = 2*n+1 by A1;
    hence a.x in NAT;
  end;
  dom a = NAT by FUNCT_2:def 1;
  hence thesis by A2,A5,FUNCT_2:3;
end;
