reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th16:
  card y=0-line = continuum
proof
  deffunc F(Real) = |[$1,0]|;
  consider f being Function such that
A1: dom f = REAL and
A2: for r being Element of REAL holds f.r = F(r) from FUNCT_1:sch 4;
  REAL, y=0-line are_equipotent
  proof
    take f;
    thus f is one-to-one
    proof
      let x,y be object;
      assume that
A3:   x in dom f and
A4:   y in dom f;
      reconsider x,y as Element of REAL by A3,A4,A1;
A5:   f.y = |[y,0]| by A2;
      f.x = |[x,0]| by A2;
      hence thesis by A5,SPPOL_2:1;
    end;
    thus dom f = REAL by A1;
    thus rng f c= y=0-line
    proof
      let a be object;
      assume a in rng f;
      then consider b being object such that
A6:   b in dom f and
A7:   a = f.b by FUNCT_1:def 3;
      reconsider b as Element of REAL by A1,A6;
      a = |[b,0]| by A2,A7;
      hence thesis;
    end;
    let a be object;
    assume
A8: a in y=0-line;
    then reconsider a as Point of TOP-REAL 2;
    reconsider a1 = a`1 as Element of REAL by XREAL_0:def 1;
A9: a = |[a`1,a`2]| by EUCLID:53;
    then a`2 = 0 by A8,Th15;
    then a = f.a1 by A2,A9;
    hence thesis by A1,FUNCT_1:def 3;
  end;
  hence thesis by CARD_1:5,TOPGEN_3:def 4;
end;
