
theorem RSC:
  for x be object holds
    x is real-valued Complex_Sequence iff x is Real_Sequence
  proof
    let x be object;
    L1: for x be Real_Sequence holds x is real-valued Complex_Sequence
    proof
      let x be Real_Sequence;
      dom x = NAT & for k be Element of NAT holds x.k is Element of COMPLEX
        by SEQ_1:2,XCMPLX_0:def 2;
      hence thesis by COMSEQ_1:2;
    end;
    for x be real-valued Complex_Sequence holds x is Real_Sequence
    proof
      let x be real-valued Complex_Sequence;
      dom x = NAT & for k be Nat holds x.k is real by COMSEQ_1:1;
      hence thesis by SEQ_1:2;
    end;
    hence thesis by L1;
  end;
