reserve c, c1, d for Real,
  k for Nat,
  n, m, N, n1, N1, N2, N3, N4, N5, M for Element of NAT,
  x for set;

theorem
  for f being eventually-nonnegative Real_Sequence, c being positive
  Real holds Big_Oh(f) = Big_Oh(c(#)f)
proof
  let f be eventually-nonnegative Real_Sequence, c be positive Real;
  now
    let x be object;
    hereby
      assume x in Big_Oh(f);
      then consider t being Element of Funcs(NAT, REAL) such that
A1:   x = t and
A2:   ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
      consider c1,N such that
A3:   c1 > 0 and
A4:   for n st n >= N holds t.n <= c1*f.n & t.n >= 0 by A2;
A5:   now
        let n;
        assume
A6:     n >= N;
        then t.n <= c1*1*f.n by A4;
        then t.n <= c1*(c"*c)*f.n by XCMPLX_0:def 7;
        then t.n <= c1*c"*(c*f.n);
        hence t.n <= c1*c"*(c(#)f).n & t.n >= 0 by A4,A6,SEQ_1:9;
      end;
      c1*c" > 0*c" by A3,XREAL_1:68;
      hence x in Big_Oh(c(#)f) by A1,A5;
    end;
    assume x in Big_Oh(c(#)f);
    then consider t being Element of Funcs(NAT, REAL) such that
A7: x = t and
A8: ex c1,N st c1 > 0 & for n st n >= N holds t.n <= c1*(c(#)f).n & t .n>= 0;
    consider c1,N such that
A9: c1 > 0 and
A10: for n st n >= N holds t.n <= c1*(c(#)f).n & t.n >= 0 by A8;
A11: now
      let n;
      assume
A12:  n >= N;
      then t.n <= c1*(c(#)f).n by A10;
      then t.n <= c1*(c*f.n) by SEQ_1:9;
      hence t.n <= (c1*c)*f.n & t.n >= 0 by A10,A12;
    end;
    c1*c > 0*c by A9,XREAL_1:68;
    hence x in Big_Oh(f) by A7,A11;
  end;
  hence thesis by TARSKI:2;
end;
