reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th14:
  Re seq1=Re seq2 & Im seq1=Im seq2 implies seq1=seq2
proof
  assume that
A1: Re seq1=Re seq2 and
A2: Im seq1=Im seq2;
  now
    let n be Element of NAT;
A3: Im(seq1.n)=Im seq2.n by A2,Def6
      .=Im(seq2.n) by Def6;
    Re(seq1.n)=Re seq2.n by A1,Def5
      .=Re(seq2.n) by Def5;
    hence seq1.n=seq2.n by A3,COMPLEX1:3;
  end;
  hence thesis;
end;
