reserve n for Nat;

theorem Th37:
  for A, B being SetSequence of the carrier of TOP-REAL 2, C, C1
  being SetSequence of the carrier of [: TOP-REAL 2, TOP-REAL 2 :] st (for i
being Nat holds C.i = [: A.i, B.i :]) & C1 is subsequence of C holds
  ex A1, B1 being SetSequence of the carrier of TOP-REAL 2 st A1 is subsequence
of A & B1 is subsequence of B & for i being Nat holds C1.i = [: A1.i
  , B1.i :]
proof
  let A, B be SetSequence of the carrier of TOP-REAL 2, C, C1 be SetSequence
  of the carrier of [: TOP-REAL 2, TOP-REAL 2 :];
  assume that
A1: for i being Nat holds C.i = [: A.i, B.i :] and
A2: C1 is subsequence of C;
  consider NS being increasing sequence of NAT such that
A3: C1 = C * NS by A2,VALUED_0:def 17;
  set B1 = B * NS;
  set A1 = A * NS;
  reconsider A1 as SetSequence of TOP-REAL 2;
  reconsider B1 as SetSequence of TOP-REAL 2;
  take A1, B1;
  for i being Nat holds C1.i = [: A1.i, B1.i :]
  proof
    let i be Nat;
A4:  i in NAT by ORDINAL1:def 12;
A5: dom NS = NAT by FUNCT_2:def 1;
    then
A6: A1.i = A.(NS.i) & B1.i = B.(NS.i) by FUNCT_1:13,A4;
    C1.i = C.(NS.i) by A3,A5,FUNCT_1:13,A4
      .= [: A1.i, B1.i :] by A1,A6;
    hence thesis;
  end;
  hence thesis;
end;
