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 Th5:
  for N,n be Nat,seq be Real_Sequence of N holds
  (r*seq).n = r*seq.n
  proof
    let N,n be Nat,seq be Real_Sequence of N;
    reconsider m = n as Element of NAT by ORDINAL1:def 12;
A1: dom(r*seq) = NAT by FUNCT_2:def 1;
    thus (r*seq).n = (r*seq)/.m
    .= r*seq/.m by A1,VFUNCT_1:def 4
    .= r*seq.n;
 end;
