reserve n for Nat;

theorem Th32:
  for A, B, C being SetSequence of the carrier of TOP-REAL 2 st (
for i being Nat holds A.i c= B.i) & C is subsequence of A holds ex D
  being subsequence of B st for i being Nat holds C.i c= D.i
proof
  let A, B, C be SetSequence of the carrier of TOP-REAL 2;
  assume that
A1: for i being Nat holds A.i c= B.i and
A2: C is subsequence of A;
  consider NS being increasing sequence of NAT such that
A3: C = A * NS by A2,VALUED_0:def 17;
  set D = B * NS;
  reconsider D as SetSequence of TOP-REAL 2;
  reconsider D as subsequence of B;
  take D;
  for i being Nat holds C.i c= D.i
  proof
    let i be Nat;
A4: dom NS = NAT by FUNCT_2:def 1;
    C.i c= D.i
    proof
      let x be object;
A5:  i in NAT by ORDINAL1:def 12;
      assume x in C.i;
      then
A6:   x in A.(NS.i) by A3,A4,FUNCT_1:13,A5;
      A.(NS.i) c= B.(NS.i) by A1;
      then x in B.(NS.i) by A6;
      hence thesis by A4,FUNCT_1:13,A5;
    end;
    hence thesis;
  end;
  hence thesis;
end;
