reserve f for Function;
reserve n,k,n1 for Nat;
reserve r,p for Real;
reserve x,y,z for object;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Real_Sequence;

theorem Th52:
  abs(seq)"=abs(seq")
proof
  now
    let n be Element of NAT;
    thus (abs(seq")).n=|.seq".n.| by Th12
      .=|.(seq.n)".| by VALUED_1:10
      .=|.1/(seq.n).| by XCMPLX_1:215
      .=1/|.seq.n.| by ABSVALUE:7
      .=(|.seq.n.|)" by XCMPLX_1:215
      .=(abs(seq).n)" by Th12
      .=(abs(seq))".n by VALUED_1:10;
  end;
  hence thesis by FUNCT_2:63;
end;
