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
  abs(r(#)seq)=|.r.|(#)abs(seq)
proof
  now
    let n be Element of NAT;
    thus abs(r(#)seq).n=|.(r(#)seq).n.| by Th12
      .=|.r*(seq.n).| by Th9
      .=|.r.|*|.seq.n.| by COMPLEX1:65
      .=|.r.|*(abs(seq)).n by Th12
      .=(|.r.|(#)abs(seq)).n by Th9;
  end;
  hence thesis by FUNCT_2:63;
end;
