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 Th50:
  abs(seq(#)seq9)=abs(seq)(#)abs(seq9)
proof
  now
    let n be Element of NAT;
    thus (abs(seq(#)seq9)).n=|.(seq(#)seq9).n.| by Th12
      .=|.(seq.n)*(seq9.n).| by Th8
      .= |.seq.n.|*|.seq9.n.| by COMPLEX1:65
      .= ((abs(seq)).n)*|.seq9.n.| by Th12
      .=((abs(seq)).n)*(abs(seq9)).n by Th12
      .=(abs(seq)(#)abs(seq9)).n by Th8;
  end;
  hence thesis by FUNCT_2:63;
end;
