reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th31:
  for u being integer-valued FinSequence
  for m being CR_Sequence st dom u = dom m holds
  { z where z is positive Nat: z solves_CRT u,m } is infinite
  proof
    let u be integer-valued FinSequence;
    let m be CR_Sequence;
    assume
A1: dom u = dom m;
    set X = { x where x is positive Nat: x solves_CRT u,m };
    CRT(u,m) solves_CRT u,m by A1,Def4;
    then CRT(u,m)+1*Product(m) solves_CRT u,m by A1,Th30;
    then
A2: CRT(u,m)+1*Product(m) in X;
A3: X is natural-membered
    proof
      let a be object;
      assume a in X;
      then ex x being positive Nat st a = x & x solves_CRT u,m;
      hence thesis;
    end;
    for a being Nat st a in X ex b being Nat st b > a & b in X
    proof
      let a be Nat;
      assume a in X;
      then consider x being positive Nat such that
A4:   a = x and
A5:   x solves_CRT u,m;
      take b = x + 1*Product(m);
      x + Product(m) > x + 0 by XREAL_1:8;
      hence b > a by A4;
      b solves_CRT u,m by A1,A5,Th30;
      hence b in X;
    end;
    hence thesis by A2,A3,NUMBER04:1;
  end;
