reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem Th12:
  r*(seq1 + seq2) = r*seq1 + r*seq2
proof
  let n be Element of NAT;
  thus (r*(seq1 + seq2)).n=r*(seq1+seq2).n by Th5
    .=r*(seq1.n+seq2.n) by Th4
    .=r*seq1.n+r*seq2.n by RLVECT_1:def 5
    .=(r*seq1).n+r*seq2.n by Th5
    .=(r*seq1).n+(r*seq2).n by Th5
    .=((r*seq1)+(r*seq2)).n by Th4;
end;
