
theorem :: Problem 54
  {<*x,y,z*> where x is Real, y is Real, z is Real :
     x*(x+1), y*(y+1), z*(z+1) form_an_AP } is infinite
  proof
    set AP = {<*x,y,z*> where x is Real, y is Real, z is Real :
       x*(x+1), y*(y+1), z*(z+1) form_an_AP };
    reconsider x = 1 as Nat;
    reconsider y = 5 * x + 2 as Nat;
    reconsider z = 7 * x + 3 as Nat;
    x * (x + 1), y * (y + 1), z * (z + 1) form_an_AP; then
    <*1,5*1+2,7*1+3*> in AP; then
    reconsider AP as non empty set;
    defpred P[Element of REAL, Element of AP] means
      $2 = <*$1,5*$1+2,7*$1+3*>;
AA: for x being Element of REAL
    ex y being Element of AP st P[x,y]
    proof
      let x be Element of REAL;
      set y = 5 * x + 2;
      set z = 7 * x + 3;
      x * (x + 1), y * (y + 1), z * (z + 1) form_an_AP; then
      <*x,5*x+2,7*x+3*> in AP; then
      reconsider yy = <*x,5*x+2,7*x+3*> as Element of AP;
      take yy;
      thus thesis;
    end;
    consider f being Function of REAL, AP such that
A0: for x being Element of REAL holds
    P[x,f.x] from FUNCT_2:sch 3(AA);
A1: dom f is infinite by FUNCT_2:def 1;
    for x1,x2 being object st
    x1 in REAL & x2 in REAL & f.x1 = f.x2 holds x1 = x2
    proof
      let x1,x2 be object;
      assume
B0:   x1 in REAL & x2 in REAL & f.x1 = f.x2; then
      reconsider xx1 = x1, xx2 = x2 as Element of REAL;
      f.xx1 = <*xx1,5*xx1+2,7*xx1+3*> by A0; then
      <*xx1,5*xx1+2,7*xx1+3*> = <*xx2,5*xx2+2,7*xx2+3*> by A0,B0;
      hence thesis by FINSEQ_1:78;
    end; then
A2: f is one-to-one by FUNCT_2:19;
    rng f c= AP by RELAT_1:def 19;
    hence thesis by A1,A2,CARD_1:59;
  end;
