
theorem :: Problem 52
  for n being Nat holds
   ex f being AP-like integer-valued XFinSequence st
     dom f >= n &
     f is with_all_coprime_terms
  proof
    let n be Nat;
    set f = ArProg (n! + 1, n!);
    reconsider ff = f | n as integer-valued XFinSequence by RestrictedXFin;
SS: for k being Nat holds
      f.k = n! * (k+1) + 1
    proof
      let k be Nat;
      f.k = n! + 1 + n! * k by ArDefNth
         .= n! * (k + 1) + 1;
      hence thesis;
    end;
S0: for k being Nat st k + 1 <= n holds
      ff.k = n! * (k+1) + 1
    proof
      let k be Nat;
      assume k + 1 <= n; then
      k < n by NAT_1:13; then
      k in Segm n by NAT_1:44; then
      ff.k = f.k by FUNCT_1:49;
      hence thesis by SS;
    end;
    dom f = NAT by FUNCT_2:def 1; then
t2: dom (f|Segm n) = Segm n;
    for i,j being Nat st i in dom ff & j in dom ff & i <> j holds
      ff.i, ff.j are_coprime
    proof
      let i,j be Nat;
      assume
S1:   i in dom ff & j in dom ff & i <> j; then
S2:   i + 1 <> j + 1;
S6:   1 <= i + 1 & 1 <= j + 1 by NAT_1:12;
      i < n & j < n by NAT_1:44,S1,t2; then
S4:   i + 1 <= n & j + 1 <= n by NAT_1:13; then
      ff.i = n! * (i + 1) + 1 & ff.j = n! * (j + 1) + 1 by S0;
      hence thesis by S4,S6,S2,LemmaFor52;
    end; then
    ff is with_all_coprime_terms;
    hence thesis by t2;
  end;
