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 Th22:
  r(#)(seq1+seq2)=r(#)seq1+r(#)seq2
proof
  now
    let n be Element of NAT;
    thus (r(#)(seq1 + seq2)).n=r*(seq1+seq2).n by Th9
      .=r*(seq1.n+seq2.n) by Th7
      .=r*seq1.n+r*seq2.n
      .=(r(#)seq1).n+r*seq2.n by Th9
      .=(r(#)seq1).n+(r(#)seq2).n by Th9
      .=((r(#)seq1)+(r(#)seq2)).n by Th7;
  end;
  hence thesis by FUNCT_2:63;
end;
