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

theorem Th67:
  for i being non zero Integer holds INT,multiples(i) are_equipotent
  proof
    let i be non zero Integer;
    set M = multiples(i);
    set I = id INT;
    set f = i(#)I;
    take f;
    thus f is one-to-one;
    thus dom f = INT by VALUED_1:def 5;
    thus rng f = M
    proof
      thus rng f c= M
      proof
        let y be object;
        assume y in rng f;
        then consider x being object such that
        x in dom f and
A1:     f.x = y by FUNCT_1:def 3;
        reconsider y as Integer by A1;
        f.x = i*I.x by VALUED_1:6;
        then i divides y by A1;
        then y is Multiple of i by Def15;
        hence thesis;
      end;
      let y be object;
      assume y in M;
      then reconsider m = y as Multiple of i by Th61;
      i divides m by Def15;
      then consider j such that
A2:   m = i*j;
      reconsider j as Element of INT by INT_1:def 2;
A3:   dom f = INT by VALUED_1:def 5;
      f.j = i*I.j by VALUED_1:6
      .= y by A2;
      hence thesis by A3,FUNCT_1:def 3;
    end;
  end;
