
theorem Th24:
  for f being CR_Sequence, m being Nat st 0 < m & m <=
  len f holds f|m is CR_Sequence
proof
  let f be CR_Sequence, m be Nat;
  reconsider fm = f|m as FinSequence of INT by FINSEQ_1:102;
  assume that
A1: m > 0 and
A2: m <= len f;
A3: len fm = m by A2,FINSEQ_1:59;
A4: now
    let i be Element of NAT;
    assume i in dom fm;
    then
A5: i in Seg m by A3,FINSEQ_1:def 3;
    then i <= m by FINSEQ_1:1;
    then
A6: i <= len f by A2,XXREAL_0:2;
    1 <= i by A5,FINSEQ_1:1;
    then i in Seg(len f) by A6;
    hence i in dom f by FINSEQ_1:def 3;
  end;
A7: now
    let i9,j9 be Nat;
    assume that
A8: i9 in dom fm and
A9: j9 in dom fm and
A10: i9 <> j9;
    reconsider i = i9,j = j9 as Element of NAT by ORDINAL1:def 12;
A11: f.i = fm.i by A8,FUNCT_1:47;
A12: f.j = fm.j by A9,FUNCT_1:47;
A13: j in dom f by A4,A9;
    i in dom f by A4,A8;
    hence fm.i9, fm.j9 are_coprime by A10,A13,A11,A12,Def2;
  end;
  now
    let r be Real;
    assume r in rng fm;
    then consider i being object such that
A14: i in dom fm and
A15: fm.i = r by FUNCT_1:def 3;
    reconsider i as Element of NAT by A14;
    f.i in rng f by A4,A14,FUNCT_1:3;
    then f.i > 0 by PARTFUN3:def 1;
    hence r > 0 by A14,A15,FUNCT_1:47;
  end;
  hence thesis by A1,A7,Def2,PARTFUN3:def 1;
end;
