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 Th4:
  for N,n be Nat,seq1,seq2 be Real_Sequence of N holds
  (seq1+seq2).n = seq1.n + seq2.n
  proof
    let N,n be Nat,seq1,seq2 be Real_Sequence of N;
    reconsider m = n as Element of NAT by ORDINAL1:def 12;
A1: dom(seq1+seq2) = NAT by FUNCT_2:def 1;
    thus (seq1+seq2).n = (seq1+seq2)/.m
    .= seq1/.m + seq2/.m by A1,VFUNCT_1:def 1
    .= seq1.n + seq2.n;
  end;
